Nuova ricerca
 Maria Rita CASALI Professore Ordinario presso: Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede ex-Matematica

## Pubblicazioni

2021 - Compact 4-manifolds admitting special handle decompositions [Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola
abstract

In this paper we study colored triangulations of compact PL $4$-manifolds with empty or connected boundary which induce handle decompositions lacking in 1-handles or in 1- and 3-handles, thus facing also the problem, posed by Kirby, of the existence of {em special handlebody decompositions} for any simply-connected closed PL $4$-manifold. In particular, we detect a class of compact simply-connected PL $4$-manifolds with empty or connected boundary, which admit such decompositions and, therefore, can be represented by (undotted) framed links. Moreover, this class includes any compact simply-connected PL $4$-manifold with empty or connected boundary having colored triangulations that minimize the combinatorially defined PL invariants {em regular genus, gem-complexity} or {em G-degree} among all such manifolds with the same second Betti number.

2020 - Classifying compact 4-manifolds via generalized regular genus and G-degree [Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola
abstract

$(d+1)$-colored graphs, i.e. edge-colored graphs that are $(d+1)$-regular, have already been proved to be a useful representation tool for compact PL $d$-manifolds, thus extending the theory (known as crystallization theory) originally developed for the closed case. In this context, combinatorially defined PL invariants play a relevant role. The present paper focuses in particular on generalized regular genus and G-degree: the first one extending to higher dimension the classical notion of Heegaard genus for 3-manifolds, the second one arising, within theoretical physics, from the theory of random tensors as an approach to quantum gravity in dimension greater than two. We establish several general results concerning the two invariants, in relation with invariants of the boundary and with the rank of the fundamental group, as well as their behaviour with respect to connected sums. We also compute both generalized regular genus and G-degree for interesting classes of compact $d$-manifolds, such as handlebodies, products of closed manifolds by the interval and $mathbb D^2$-bundles over $mathbb S^2.$ The main results of the paper concern dimension 4, where it is obtained the classification of all compact PL manifolds with generalized regular genus at most one, and of all compact PL manifolds with G-degree at most 18; moreover, in case of empty or connected boundary, the classifications are extended to generalized regular genus two and to G-degree 24.

2020 - Crystallizations of compact 4-manifolds minimizing combinatorially defined PL-invariants [Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola; Gagliardi, Carlo
abstract

The present paper is devoted to present a unifying survey about some special classes of crystallizations of compact PL $4$-manifolds with empty or connected boundary, called semi-simple and weak semi-simple crystallizations, with a particular attention to their properties of minimizing combinatorially defined PL-invariants, such as the regular genus, the Gurau degree, the gem-complexity and the (gem-induced) trisection genus. The main theorem, yielding a summarizing result on the topic, is an original contribution. Moreover, in the present paper the additivity of regular genus with respect to connected sum is proved to hold for all compact $4$-manifolds with empty or connected boundary which admit weak semi-simple crystallizations.

2019 - Combinatorial properties of the G-degree [Articolo su rivista]
Casali, M. R.; Grasselli, L.
abstract

A strong interaction is known to exist between edge-colored graphs (which encode PL pseudo-manifolds of arbitrary dimension) and random tensor models (as a possible approach to the study of Quantum Gravity). The key tool is the "G-degree" of the involved graphs, which drives the 1/N expansion in the tensor models context. In the present paper - by making use of combinatorial properties concerning Hamiltonian decompositions of the complete graph - we prove that, in any even dimension d greater or equal to 4, the G-degree of all bipartite graphs, as well as of all (bipartite or non-bipartite) graphs representing singular manifolds, is an integer multiple of (d-1)!. As a consequence, in even dimension, the terms of the 1/N expansion corresponding to odd powers of 1/N are null in the complex context, and do not involve colored graphs representing singular manifolds in the real context. In particular, in the 4-dimensional case, where the G-degree is shown to depend only on the regular genera with respect to an arbitrary pair of "associated" cyclic permutations, several results are obtained, relating the G-degree or the regular genus of 5-colored graphs and the Euler characteristic of the associated PL 4-manifolds.

2018 - G-degree for singular manifolds [Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola; Grasselli, Luigi
abstract

The G-degree of colored graphs is a key concept in the approach to Quantum Gravity via tensor models. The present paper studies the properties of the G-degree for the large class of graphs representing singular manifolds (including closed PL manifolds). In particular, the complete topological classification up to G-degree 6 is obtained in dimension 3, where all 4-colored graphs represent singular manifolds.

2018 - TOPOLOGY IN COLORED TENSOR MODELS [Poster]
Casali, M. R.; Cristofori, P.; Grasselli, L.
abstract

From a “geometric topology” point of view, the theory of manifold representation by means of edge-colored graphs has been deeply studied since 1975 and many results have been achieved: its great advantage is the possibility of encoding, in any dimension, every PL d-manifold by means of a totally combinatorial tool. Edge-colored graphs also play an important rôle within colored tensor models theory, considered as a possible approach to the study of Quantum Gravity: the key tool is the G-degree of the involved graphs, which drives the 1/N expansion in the higher dimensional tensor models context, exactly as it happens for the genus of surfaces in the two-dimensional matrix model setting. Therefore, topological and geometrical properties of the represented PL manifolds, with respect to the G-degree, have specific relevance in the tensor models framework, show- ing a direct fruitful interaction between tensor models and discrete geometry, via edge-colored graphs. In colored tensor models, manifolds and pseudomanifolds are (almost) on the same footing, since they constitute the class of polyhedra represented by edge-colored Feynman graphs arising in this context; thus, a promising research trend is to look for classification results concerning all pseudomanifolds - or, at least, singular d-manifolds, if d ≥ 4 - represented by graphs of a given G-degree. In dimension 4, the existence of colored graphs encoding different PL manifolds with the same underlying TOP manifold, suggests also to investigate the ability of ten- sor models to accurately reflect geometric degrees of freedom of Quantum Gravity.

2018 - Topology in colored tensor models via crystallization theory [Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola; Dartois, Stèphane; Grasselli, Luigi
abstract

The aim of this paper is twofold. On the one hand, it provides a review of the links between random tensor models, seen as quantum gravity theories, and the PL-manifolds representation by means of edge-colored graphs (crystallization theory). On the other hand, the core of the paper is to establish results about the topological and geometrical properties of the Gurau-degree (or G-degree) of the represented manifolds, in relation with the motivations coming from physics. In fact, the G-degree appears naturally in higher dimensional tensor models as the quantity driving their 1/N expansion, exactly as it happens for the genus of surfaces in the two-dimensional matrix model setting. In particular, the G-degree of PL-manifolds is proved to be finite-to-one in any dimension, while in dimension 3 and 4 a series of classification theorems are obtained for PL-manifolds represented by graphs with a fixed G-degree. All these properties have specific relevance in the tensor models framework, showing a direct fruitful interaction between tensor models and discrete geometry, via crystallization theory.

2017 - Lower bounds for regular genus and gem-complexity of PL 4-manifolds [Articolo su rivista]
Basak, B.; Casali, Maria Rita
abstract

Within crystallization theory, two interesting PL invariants for d-manifolds have been introduced and studied, namely, gem-complexity and regular genus. In the present paper we prove that for any closed connected PL 4-manifold M, its gem-complexity k(M) and its regular genus G(M) satisfy k(M)≥3χ(M)+10m−6 and G(M)≥2χ(M)+5m−4, where rk(π1(M))=m. These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of product 4-manifolds. Moreover, the class of semi-simple crystallizations is introduced, so that the represented PL 4-manifolds attain the above lower bounds. The additivity of both gem-complexity and regular genus with respect to connected sum is also proved for such a class of PL 4-manifolds, which comprehends all ones of “standard type”, involved in existing crystallization catalogs, and their connected sums.

2016 - Classifying PL 4-manifolds via crystallizations: results and open problems [Capitolo/Saggio]
Casali, Maria Rita; Cristofori, Paola; Gagliardi, Carlo
abstract

Crystallization theory is a graph-theoretical representation method for compact PL-manifolds of arbitrary dimension, which makes use of a particular class of edge-coloured graphs, which are dual to coloured (pseudo-)triangulations. The purely combinatorial nature of crystallizations makes them particularly suitable for automatic generation and classication, as well as for the introduction and study of graph-defined invariants for PL-manifolds. The present survey paper focuses on the 4-dimensional case, presenting up-to-date results about the PL classication of closed 4-manifolds, by means of two such PL invariants: regular genus and gem-complexity. Open problems are also presented, mainly concerning different classication of 4-manifolds in TOP and DIFF=PL categories, and a possible approach to the 4-dimensional Smooth Poincare Conjecture.

2016 - GEOMETRIA [Monografia/Trattato scientifico]
Casali, Maria Rita; Gagliardi, Carlo; Grasselli, Luigi
abstract

Il presente testo sviluppa argomenti tradizionalmente trattati nei corsi di “Geometria” (ovvero di “Algebra e Geometria”) nell'ambito delle lauree di primo livello, ed è particolarmente rivolto agli studenti dei vari Corsi di Laurea in Ingegneria, e di quelli in Matematica, Fisica ed Informatica. Il testo è suddiviso in due parti: - la prima parte contiene gli elementi fondamentali di Algebra lineare; - la seconda, di carattere più propriamente geometrico, riguarda le principali proprietà degli spazi euclidei, sviluppando in tale ambito la teoria delle coniche e delle quadriche. L'esposizione risulta articolata, come ovvio per ogni teoria matematica, in Definizioni e Proposizioni (o Teoremi, nel caso in cui gli enunciati rivestano particolare importanza). Un ruolo significativo viene attribuito a Osservazioni ed Esempi atti a: - chiarire concetti, risultati, dimostrazioni; - stimolare i necessari collegamenti tra i vari argomenti; - motivare la genesi dei concetti e dei problemi; - evidenziare i casi notevoli di particolare rilievo nell'ambito di una teoria generale; - indicare possibili generalizzazioni o descrizioni alternative di una teoria. Ciò può consentire inoltre al Docente di “dosare” con maggiore libertà, secondo le proprie convinzioni ed esperienze didattiche, il peso da attribuire, durante le lezioni, ai vari argomenti del corso. Definizioni, Proposizioni e Osservazioni sono dotati di una numerazione progressiva all'interno di ogni Capitolo; l'esposizione della teoria è arricchita inoltre da esempi notevoli, con numerazione autonoma all'interno di ogni Capitolo. Con l'eccezione delle principali proprietà degli insiemi numerici fondamentali e dell'utilizzo di una teoria “ingenua”, non rigorosamente assiomatica, degli insiemi (peraltro, brevemente richiamata nel primo Capitolo), il testo appare essenzialmente autocontenuto. In particolare, non risulta necessario alcun prerequisito di Geometria euclidea così come viene sviluppata, a partire da un sistema di assiomi, nelle Scuole secondarie. Seguendo l'impostazione algebrica ormai dominante nelle varie teorie matematiche e quindi in una ottica di “algebrizzazione della Geometria”, i concetti e i risultati di natura geometrica, compresi quelli relativi alla Geometria euclidea, sono infatti ricavati da conoscenze di tipo algebrico precedentemente introdotte. Abbiamo cercato tuttavia di non fare perdere contenuto geometrico a tali concetti, sia mediante il metodo con cui questi vengono presentati, sia facendo spesso ricorso ad Osservazioni ed Esempi atti ad aiutare il lettore a ritrovare, pure in ambiti più generali, le proprietà geometriche già note. La scelta privilegiata è stata quella di sviluppare la teoria, sia dal punto di vista algebrico che da quello geometrico, per spazi di dimensione finita n; le dimensioni due e tre sono tuttavia sempre illustrate in modo dettagliato, come casi particolari e nelle loro specificità, sfruttandone le caratteristiche di rappresentatività. Tale scelta di generalità nella dimensione è dovuta essenzialmente a due considerazioni: da un lato riteniamo opportuno evitare inutili ripetizioni nella enunciazione della teoria per le varie dimensioni particolari, dall'altro siamo convinti che lo sviluppo della teoria in ambito ragionevolmente generale sia un ottimo stimolo allo sviluppo della capacità di astrazione e generalizzazione che è obiettivo fondamentale di ogni corso di matematica, anche nell'ambito dei nuovi ordinamenti degli studi universitari. La presente III edizione risulta integrata, rispetto a quelle precedenti, in primo luogo con l'introduzione di test di valutazione al termine di ciascuna delle due parti (Algebra lineare e Geometria euclidea) in cui il testo è suddiviso, rendendo così possibile al lettore una verifica del proprio livello di comprensione delle tematiche trattate. È stata inoltre realizzata una rivisitazione sostanziale di alcuni argomenti (in pa

2016 - PL 4-manifolds admitting simple crystallizations: framed links and regular genus [Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola; Gagliardi, Carlo
abstract

Simple crystallizations are edge-colored graphs representing piecewise linear (PL) 4-manifolds with the property that the 1-skeleton of the associated triangulation equals the 1-skeleton of a 4-simplex. In this paper, we prove that any (simply-connected) PL 4-manifold M admitting a simple crystallization admits a special handlebody decomposition, too; equivalently, M may be represented by a framed link yielding S^3, with exactly β_2(M) components (β_2(M) being the second Betti number of M). As a consequence, the regular genus of M is proved to be the double of β_2(M). Moreover, the characterization of any such PL 4-manifold by k(M)=3β_2(M), where k(M) is the gem-complexity of M (i.e. the non-negative number p−1, 2p being the minimum order of a crystallization of M), implies that both PL invariants gem-complexity and regular genus turn out to be additive within the class of all PL 4-manifolds admitting simple crystallizations (in particular, within the class of all “standard” simply-connected PL 4-manifolds).

2015 - A note about complexity of lens spaces [Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola
abstract

Within crystallization theory, (Matveev's) complexity of a 3-manifold can be estimated by means of the combinatorial notion of GM-complexity. In this paper, we prove that the GM-complexity of any lens space L(p,q), with p greater than 2, is bounded by S(p,q)-3, where S(p,q) denotes the sum of all partial quotients in the expansion of q/p as a regular continued fraction. The above upper bound had been already established with regard to complexity; its sharpness was conjectured by Matveev himself and has been recently proved for some infinite families of lens spaces by Jaco, Rubinstein and Tillmann. As a consequence, infinite classes of 3-manifolds turn out to exist, where complexity and GM-complexity coincide. Moreover, we present and briefly analyze results arising from crystallization catalogues up to order 32, which prompt us to conjecture, for any lens space L(p,q) with p greater than 2, the following relation: k(L(p,q)) = 5 + 2 c(L(p,q)), where c(M) denotes the complexity of a 3-manifold M and k(M)+1 is half the minimum order of a crystallization of M

2015 - Cataloguing PL 4-manifolds by gem-complexity [Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola
abstract

We describe an algorithm to subdivide automatically a given set of PL n-manifolds (via coloured triangulations or, equivalently, via crystallizations) into classes whose elements are PL-homeomorphic. The algorithm, implemented in the case n = 4, succeeds to solve completely the PL-homeomorphism problem among the catalogue of all closed connected PL 4-manifolds up to gem-complexity 8 (i.e., which admit a coloured triangulation with at most 18 4-simplices). Possible interactions with the (not completely known) relationship among the different classications in the TOP and DIFF=PL categories are also investigated. As a first consequence of the above PL classification, the non-existence of exotic PL 4-manifolds up to gem-complexity 8 is proved. Further applications of the tool are described, related to possible PL-recognition of different triangulations of the K3-surface.

2014 - REVIEW OF: "Benedetti Riccardo - Petronio Carlo, Spin structures on 3-manifolds via arbitrary triangulations, Algebr. Geom. Topol. 14, No. 2, 1005-1054 (2014)". [DE062726187] [Recensione in Rivista]
Casali, Maria Rita
abstract

In the present paper a combinatorial encoding of spin structures based on arbitrary triangulations of oriented compact 3-manifolds is introduced. The goal is achieved by means of the notion of weak branching, which turns out to be related to the notion of Z/2Z-taut structure on triangulations, introduced by Luo (see [Proc. Am. Math. Soc. 140, No. 3, 1053-1068 (2012; Zbl 1250.57028)] and [Proc. Am. Math. Soc. 141, No. 1, 335-350 (2013; Zbl 1272.57004)]). In particular, by taking into account the set of all pairs (M, s) (M being a compact oriented 3-manifold and s being a spin structure on M), the authors claim: - “given any (loose) triangulation T of M, with ideal vertices at the components of @M and possibly internal vertices, and any s, we encode s by decorating T with certain extra combinatorial structures; - we exibit combinatorial moves on decorated triangulations relating to each other any two that encode the same (M, s).” A dual version of the above encoding is also presented, in terms of special spines dual to triangulations (see [Acta Appl. Math. 19, No.2, 101-130 (1990; Zbl 0724.57012)]). A first application of the described techniques is contained in [Baseilhac-Benedetti, Analytic families of quantum hyperbolic invariants and their asymptotical behaviour, I, arXiv:1212.4261]. Further possible applications are also pointed out, concerning “an effective construction of the Roberts spin-refined TuraevViro invariants and of the related Blanchet spin-refined ReshetikhinTuraev invariants of the double of a manifold”.

The definition of Matveev complexity c(M) of a compact 3-manifold with nonempty boundary M is based on the existence of an almost simple spine for M: see [Acta Appl. Math. 19 (2), 101130 (1990; Zbl 0724.57012)]. The complexity of a given 3-manifold is generally hard to compute from the theoretical point of view, leaving aside the concrete enumeration of its spines: see, for example, [ACM monogr. 9 (2003; Zbl 1048.57001)] and [Algebr. Geom. Topol. 11 (3), 1257-1265 (2011; Zbl. 1229.57010)], together with their references. In the paper under discussion the authors establish an upper bound for the Matveev complexity of any Seifert fibered 3-manifold with nonempty boundary M, by realizing M as an assembling of several copies of five particular building blocks, whose skeleta contain a known number of true vertices. As a consequence, they obtain potentially sharp bounds on the Matveev complexity of torus knot complements.

In [Osaka J. Math. 31(3), 489-495 (1994; Zbl 0849.57018 )], the author stated that every closed connected orientable 4-manifold M with infinite cyclic fundamental group is TOP-split, i.e. it is homeomorphic to the connected sum (S1 × S3)#M1, M1 being a closed simply connected 4-manifold. However, in [Manuscr. Math. 93(4), 435-442 (1997; Zbl 0890.57034)], Hambleton and Teichner obtained a counterexample to the above general statement. In the paper under review, the author makes a revision and proves that TOP-splittability holds under the additional hypothesis that a finite covering of M is TOP-split. In particular, the original statement turns out to be true in the case of indefinite intersection form, as well as for any smooth spin 4-manifold (with infinite cyclic fundamental group). The proof of the revised statement makes use of notions developed in [Knots in Hellas 98, Ser. Knots Everything. 24 (World Scientific Publishing), 208-228 (2000; Zbl 0969.57020)] and [Atti Semin. Mat. Fis. Univ. Modena 48(2), 405-424 (2000; Zbl 1028.57019)], together with the key result - proved in [Osaka J. Math. 31(3), 489-495 (1994; Zbl 0849.57018 )] - that every closed connected orientable 4-manifold M with infinite cyclic fundamental group is homology cobordant to (S1 × S3)#M1. Consequences about surface-knots in S4 are also considered (see [J. Knot Theory Ramifications 4(2), 213-224 (1995; Zbl 0844.57020)]).

2014 - REVIEW OF: "Taylor Scott A., Band-taut sutured manifolds, Algebr. Geom. Topol. 14, No. 1, 157-215 (2014)". [DE062342080] [Recensione in Rivista]
Casali, Maria Rita
abstract

A well known theorem of Lackenby ([Math. Ann. 308, No.4, 615-632 (1997; Zbl 0876.57015)]) relates Dehn surgery properties of a knot to the intersection between the knot and essential surfaces in the 3-manifold. In the paper under review, the author extends Lackenby’s Theorem to the case of 2-handles attached to a sutured 3-manifold along a suture, by determining the relationship between an essential surface in a sutured 3-manifold, the number of intersections between the boundary of the surface and one of the sutures, and the cocore of the 2-handle in the manifold after attaching a 2-handle along the suture. The author makes use of Scharlemann’s combinatorial version of sutured manifold theory ([J. Differ. Geom. 29, No.3, 557-614 (1989; Zbl 0673.57015)]) and takes inspiration from Gabai’s proof that, under suitable hypotheses, there is at most one way to fill a torus boundary component of a 3-manifold so that the Thurston norm decreases ([J. Differ. Geom. 26, 461-478 (1987; Zbl 0627.57012)]). On the other hand, in order to prove the theorem, band-taut sutured manifolds are introduced and band-taut sutured manifold hierarchies are proved to exist. As an application, the paper shows that tunnels for tunnel number one knots or links in any 3-manifold can be isotoped to lie on a branched surface corresponding to a certain taut sutured manifold hierarchy of the knot or link exterior. Other interesting applications are contained in [Trans. Am. Math. Soc. 366, No. 7, 3747-3769 (2014; Zbl 06303179)], where band sums are proved to satisfy the cabling conjecture, and new proofs that unknotting number one knots are prime and that genus is superadditive under band sum are obtained.

2013 - Coloured graphs representing PL 4-manifolds [Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola
abstract

Crystallization theory is a representation method for compact PL manifolds by means of a particular class of edge-coloured graphs. The combinatorial nature of this representation allows to elaborate and implement algorithmic procedures for the generation and analysis of catalogues of closed PL n-manifolds. In this paper we discuss the concepts which are involved in these procedures for n = 4 and present classification results arising from the study of the initial segment of the catalogue.

2013 - Computing Matveev's complexity via crystallization theory: The boundary case [Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola
abstract

The notion of Gem-Matveev complexity (GM-complexity) has been introduced within crystallization theory, as a combinatorial method to estimate Matveev's complexity of closed 3-manifolds; it yielded upper bounds for interesting classes of such manifolds. In this paper, we extend the definition to the case of non-empty boundary and prove that for each compact irreducible and boundary-irreducible 3-manifold it coincides with the modified Heegaard complexity introduced by Cattabriga, Mulazzani and Vesnin. Moreover, via GM-complexity, we obtain an estimation of Matveev's complexity for all Seifert 3-manifolds with base D2 and two exceptional fibers and, therefore, for all torus knot complements.

2013 - Gamma-class_4dim: A program to subdivide a set of rigid crystallizations of closed 4-manifolds into equivalence classes, whose elements represent PL-homeomorphic manifolds. [Software]
Casali, Maria Rita; Cristofori, Paola
abstract

Gamma-class_4dim is a program yielding, from any given list X of crystallizations of 4-dimensional PL-manifolds, the automatic partition of the elements of X into equivalence classes, such that each class contains only crystallizations representing the same PL-manifold. Moreover, the program attempts the identification of the represented 4-manifolds by means of comparison of the representatives of each class with known catalogues of crystallizations and/or by means of splitting into connected sums. Gamma-class_4dim is based on the existence of elementary combinatorial moves available for crystallizations of PL-manifolds of any dimension (i.e. the well-known "dipole moves", together with the so called "blobs" and "flips", introduced in [S. Lins - M. Mulazzani, Blobs and flips on gems, Journal of Knot Theory and its Ramifications 15 (2006), 1001-1035]. The program has already been tested for known catalogues of crystallizations of 4-manifolds, by making use of a fixed admissible sequence of the above moves; further applications are in progress.

2013 - REVIEW OF: "Birman Joan - Brinkmann Peter - Kawamuro Keiko, Polynomial invariants of pseudo-Anosov maps, J. Topol. Anal. 4, No. 1, 13-47 (2012)". [DE060376660] [Recensione in Rivista]
Casali, Maria Rita
abstract

In [Topology 34, No.1, 109-140 (1995; Zbl 0837.57010)], Bestvina and Handel gave an algorithmic proof of Thurston’s classification theorem for mapping classes (see e.g., [Astrisque, 66-67 (1979; Zbl 0446.57005-23)]). If [F] is a pseudo-Anisov map acting on an orientable surface S, their algorithm allows to construct a graph G (homotopic to S when S is punctured), a suitable map f : G ! G (called train track map) and the associated transition matrix T (whose Perron-Frobenius eigenvalue is the dilatation of [F]: see [The theory of matrices, Vol. 2, AMS Chelsea Publishing (1959; Zbl 0927.15002)]). The dilatation (F) is an invariant of the conjugacy class [F] in the modular group of S, studied in [Ann. Sci. c. Norm. Supr. (4) 33, No. 4, 519-560 (2000; Zbl 1013.57010)] and in several subsequent papers. The present paper introduces a new approach to the study of invariants of [F], when [F] is pseudo-Anisov: starting from Bestvina-Handel algorithm, the authors investigate the structure of the characteristic polynomial of the transition matrix T and obtain two new integer polynomials (both containing (F) as their largest real root), which turn out to be invariants of the given pseudo-Anisov mapping class. The degrees of these new polynomials, as well as of their product, are invariants of [F], too; simple formulas are given for computing them by a counting argument from an invariant train track. The paper gives also examples of genus 2 pseudo-Anisov maps having the same dilatation, which are distinguished by the new invariants.

2013 - REVIEW OF: "Jaco William - Rubinstein J.Hyam - Tillmann Stephan, Z2-Thurston norm and complexity of 3-manifolds, Math. Ann. 356, No. 1, 1-22 (2013)". [DE061684636] [Recensione in Rivista]
Casali, Maria Rita
abstract

Given a closed, irreducible 3-manifold M (different from S3, RP2 and L(3, 1)), the complexity of M is known to be the minimum number of tetrahedra in a (pseudo-simplicial) triangulation of M: see [Acta Appl. Math. 19, No.2, 101-130 (1990; Zbl 0724.57012)] and [Algorithms and Computation in Mathematics 9, Springer (2007; Zbl 1128.57001)]. In [Algebr. Geom. Topol. 11(3), 1257-1265 (2011; Zbl 1229.57010)] the authors found a lower bound for the complexity of M, in case M having a connected double cover (or, equivalently, a non-trivial Z2-cohomology class). The present paper makes use of the notion of Z2-Thurston norm (an analogue of Thurston’s norm, defined in [Mem. Am. Math. Soc. 339, 99-130 (1986; Zbl 0585.57006)]) in order to obtain a new lower bound on the complexity of M, if M admits multiple Z2-cohomology classes. Moreover, the minimal triangulations realizing this bound are characterized, in terms of normal surfaces consisting entirely of quadrilateral discs. It is worthwhile to note that the combinatorial structure of a minimal triangulation turns out to be governed by 0-efficiency ([J. Differ. Geom. 65(1), 61-168 (2003; Zbl 1068.57023)]) and low degree edges ([J. Topol. 2(1), 157-180 (2009; Zbl 1227.57026)]), and that the unique minimal triangulation of the generalized quaternionic space S3/Q8k (8k 2 Z+) - already obtained by the authors via the first lower bound - actually realizes this new bound, too.

2013 - REVIEW OF: "Kotschick D. - Neofytidis, C., On three-manifolds dominated by circle bundles, Math. Z. 274, No. 1-2, 21-32 (2013)". [DE06176500X] [Recensione in Rivista]
Casali, Maria Rita
abstract

Given two closed oriented n-manifolds M and N, M is said to dominate N if a non-zero degree map from M to N exists. From dimension n = 3 on, the domination relation fails to be an ordering. By a result of [Math. Semin. Notes, Kobe Univ. 9, 159-180 (1981; Zbl 0483.57003)], every 3-manifold turns out to be dominated by a surface bundle over the circle; on the other hand, in [J. Lond. Math. Soc. 79 (3), 545-561 (2009; Zbl 1168.53024)] and [Groups Geom. Dyn. 7 (1), 181-204 (2013; Zbl 06147449)] it is shown that 3-manifolds dominated by products cannot have hyperbolic or Sol3-geometry, and must often be prime. In the present paper, the authors give a complete characterization of 3-manifolds dominated by products, by proving that a closed oriented 3-manifold is dominated by a product if and only if it is finitely covered either by a product or by a connected sum of copies of S2 × S1. It is worthwhile to note that the same characterization may also be formulated in terms of Thurston geometries, or in terms of purely algebraic properties of the fundamental group. Moreover, the authors determine which 3-manifolds are dominated by non-trivial circle bundles, and which 3-manifold groups are presentable by products (according to [J. Lond. Math. Soc. 79 (3), 545-561 (2009; Zbl 1168.53024)]).

2013 - REVIEW OF: "Li Tao, Rank and genus of 3-manifolds, J. Am. Math. Soc. 26, No. 3, 777-829 (2013)". [DE061686069] [Recensione in Rivista]
Casali, Maria Rita
abstract

In the 1960’s,Waldhausen conjectured the equality, for each closed orientable 3-manifoldM, between the rank r(M) of the fundamental group 1(M) and the Heegaard genus g(M) of M ([Algebr. Geom. Topol. 32(2), 313-322 (1978; Zbl 0397.57007)]). Various counterexamples may be found in the literature: in particular, Boileau and Zieschang obtained a Seifert fibered space with r(M) = 2 and g(M) = 3 ([Invent. Math. 76, 455-468 (1984; Zbl 0538.57004)]), while Schultens and Weidman proved the existence of graph manifolds with discrepancy g(M) − r(M) arbitrarily large ([Pac. J. Math. 231 (2), 481-510 (2007; Zbl 1171.57020)]). In 2007, the above Conjecture has been re-formulated, by restricting the attention to hyperbolic 3-manifolds: see [Geometry and Topology Monographs 12, 335-349 (2007; Zbl 1140.57009)]. The present paper gives a negative answer to this “modern version” of Waldhausen Conjecture. In fact, a closed orientable hyperbolic 3-manifold M is proved to exist, so that g(M) − r(M) is arbitrarily large. Actually, the author produces an (atoroidal) 3-manifold with boundary ¯M with r( ¯M ) < g( ¯M), and the closed counterexample is constructed starting from ¯M , via the so called annulus sum (see [Geom. Topol. 14 (4), 1871-1919 (2010; Zbl 1207.57031)]). Moreover, every 2-bridge knot exterior is proved to be a JSJ piece of a closed 3-manifold M with r(M) < g(M).

2012 - Catalogues of PL-manifolds and complexity estimations via crystallization theory [Articolo su rivista]
Casali, Maria Rita
abstract

Crystallization theory is a graph-theoretical representation method for compact PL-manifolds of arbitrary dimension, with or without boundary, which makes use of a particular class of edge-coloured graphs, which are dual to coloured (pseudo-) triangulations. These graphs are usually called gems, i.e. Graphs Encoding Manifolds, or crystallizations if the associated triangulation has the minimal number of vertices.One of the principal features of crystallization theory relies on the purely combinatorial nature of the representing objects, which makes them particularly suitable for computer manipulation.The present talk focuses on up-to-date results about:- generation of catalogues of PL-manifolds for increasing values of the vertex number of the representing graphs;- definition and/or computation of invariants for PL-manifolds, directly from the representing graphs.

2012 - Complexity computation for compact 3-manifolds via crystallizations and Heegaard diagrams [Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola; M., Mulazzani
abstract

The idea of computing Matveev complexity by using Heegaard decompositions has been recently developed by two different approaches: the first one for closed 3-manifolds via crystallization theory, yielding the notion of Gem-Matveev complexity; the other one for compact orientable 3-manifolds via generalized Heegaard diagrams, yielding the notion of modified Heegaard complexity. In this paper we extend to the non-orientable case the definition of modified Heegaard complexity and prove that for closed 3-manifolds Gem-Matveev complexity and modified Heegaard complexity coincide. Hence, they turn out to be useful different tools to compute the same upper bound for Matveev complexity.

The present paper deals about the use of normal surface theory in 3-dimensional computational topology: see Kneser’s foundational paper ([Jahresbericht D. M. V. 38, 248-260 (1929; JFM 55.0311.03)]), together with the developments by Haken and Jago-Oertel ([Acta Math. 105, 245-375 (1961; Zbl 0100.19402)], [Math. Z. 80, 89-120 (1962; Zbl 0106.16605)], [Topology 23, 195-209 (1984; Zbl 0545.57003)]). In particular, the author faces the crucial problem of enumerating normal surfaces in a given (triangulated) 3-manifold, via the underlying procedure of enumerating admissible vertices of a high-dimensional polytope (admissibility being a powerful but non-linear and non-convex constraint). The main results of the present paper are significant improvements upon the best known asymptotic bounds on the number of admissible vertices (see [J. ACM 46, No. 2, 185-211 (1999; Zbl 1065.68667)], [B.A.Burton, The complexity of the normal surface solution space, in: SCG’10: Proceedings of the Twenty-Sixth Annual Symposium on Computational Geometry, ACM Press, 2010, pp.201-209] and [Math. Comput. 79, No. 269, 453-484 (2010; Zbl pre05776230)]). To achieve these results, the author examines the layout of admissible points within polytopes in both the standard normal surface coordinate system and the streamlined quadrilateral coordinate system. These points are proved to correspond to well-behaved substructures of the face lattice, and the properties of the corresponding “admissible faces” are studied. Key lemmata include upper bounds on the number of maximal admissible faces of each dimension, and a bijection between the maximal admissible faces in the two coordinate systems mentioned above.

2012 - REVIEW of: "Friedl Stefan - Vidussi, Stefano, Twisted Alexander polynomials detect fibered 3-manifolds, Ann. Math. (2) 173, No. 3, 1587-1643 (2011)." [DE05960690X] [Recensione in Rivista]
Casali, Maria Rita
abstract

It is well known that, if a knot K S3 is fibered, then the Alexander polynomial is monic and the degree equals twice the genus of K. Various generalizations of this result have been performed, showing that twisted Alexander polynomials give necessary conditions for (N, ) (where N is a compact, connected, oriented 3- manifolds with empty or toroidal boundary and 2 H1(N;Z)) to fiber: see [Ann. Sci. Ecol. Norm. Super. (4) 35, No. 2, 153-171 (2002; Zbl 1009.57021)], [Trans. Am. Math. Soc. 355, No.10, 4187-4200 (2003; Zbl 1028.57004)], [Comment. Math. Helv. 80, No. 1, 51-61 (2005; Zbl 1066.57008)], [Topology 45, No. 6, 929-953 (2006; Zbl 1105.57009)] and [T.Kitayama, Normalization of twisted Alexander invariants, preprint 2007 (arXiv 0705.2371)]. In general, the constraint of monicness and degree for the ordinary Alexander polynomial falls short from characterizing fibered 3-manifolds. The main result of present paper shows that that the collection of all twisted Alexander polynomials does detect fiberedness; equivalently, it proves that twisted Alexander polynomials detect whether (N, ) fibers under the assumption that the Thurston norm of is known. Moreover, by making use of some of their previous works (see in particular [Am. J. Math. 130, No. 2, 455- 484 (2008; Zbl 1154.57021)]), the authors show that, if a manifold of the form S1 × N3 admits a symplectic structure, then N fibers over S1.

A sequence of rational functions in a variable q is q-holonomic (see [J. Comput. Appl. Math. 32, No.3, 321-368 (1990; Zbl 0738.33001)] and [Invent. Math. 103, No.3, 575-634 (1991; Zbl 0739.05007)]) if it satisfies a linear recursion with coefficient polynomials in q and qn. In virtue of a fundamental result by Wilf-Zeilberger, Quantum Topology turns out to provide us with a plethora of q-holonomic sequences of natural origin. In particular, the present paper takes into account the q-holonomic sequence of Jones polynomials of a knot and its parallels (see [Geom. Topol. 9, 1253-1293 (2005; Zbl 1078.57012)]). The author associates a tropical curve (see [Contemporary Mathematics 377, 289-317 (2005; Zbl 1093.14080)] and [Math. Mag. 82, No. 3, 163-173 (2009; Zbl 1227.14051)]) to each q-holonomic sequence; in particular, to every knot K a tropical curve is associated, via the Jones polynomial of K and its parallels. As a consequence, a relation is established between the AJ Conjecture ([Geometry and Topology Monographs 7, 291-309 (2004; Zbl 1080.57014)]) and the Slope Conjecture ([Quantum Topol. 2, No. 1, 43-69 (2011; Zbl 1228.57004)]), which relate the Jones polynomial of K and its parallels respectively to the SL(2;C) character variety and to slopes of incompressible surfaces. The paper gives also an explicit computation of the tropical curve for the 41, 52 and 61 knots, verifying in these cases the duality between the tropical curve and a Newton subdivision of the A-polynomial of the knot.

2012 - REVIEW OF: "Schleimer Saul, The end of the curve complex, Groups Geom. Dyn. 5, No. 1, 169-176 (2011)". [DE059733321] [Recensione in Rivista]
Casali, Maria Rita
abstract

If S is a genus g surface with b boundary components, so that 3g − 3 + b 2, then the curve complex C(S) has a vertex for each isotopy class of essential non-peripheral simple closed curves in S and a k-simplex for each collection of k + 1 disjoint vertices having disjoint representatives. By regarding each simplex as a Euclidean simplex of side-length one, C(S) turns out to be Gromov hyperbolic ([Invent. Math. 138, No.1, 103-149 (1999; Zbl 0941.32012)]). The present paper proves that, if the surface S has exactly one boundary component and genus two or more, than for each vertex ! 2 C(S) and for any r 2 N, the subcomplex spanned by C0(S) − B(!, r) is connected (where B(!, r) denotes the ball of radius r about the vertex !). In order to prove the above result, the author makes use of the fact that the complex of curves has no dead ends (Prop. 3.1 of this paper) and of the so called Birman short exact sequence (see [Annals of Mathematics Studies 82, Princeton (1975; Zbl 0305.57013)] and [Acta Math. 146, 231-270 (1981; Zbl 0477.32024)]). Note that, for the considered surfaces, the above result directly answers a question of Masur’s, and answers a question of G.Bell and K.Fujiara ([J. Lond. Math. Soc., II. Ser. 77, No. 1, 33-50 (2008; Zbl 1135.57010)]) in the negative. It is also evidence for a positive answer to a question of P.Storm (already verified in an independent way by Gabai in [Geom. Topol. 13, No. 2, 1017-1041 (2009; Zbl 1165.57015)]).

2012 - REVIEW of: "Wong Helen, Quantum invariants can provide sharp Heegaard genus bounds, Osaka J. Math. 48, No. 3 (2011), 709-717". [DE059690467] [Recensione in Rivista]
Casali, Maria Rita
abstract

It is well-known that the Heegaard genus g(M) of a 3-manifold M (i.e. the smallest integer so that M has a Heegaard splitting of that genus) is generally very difficult to compute. The present paper investigates the effectiveness of a lower bound on g(M) deriving from the Reshetikhin-Turaev invariants (see [Commun. Math. Phys. 121, No.3, 351-399 (1989; Zbl 0667.57005)] for a basic reference about the invariants, and [Topology 37, No.1, 219-224 (1998; Zbl 0892.57005)] and [Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics 18, Walter de Gruyter, Berlin (1994; Zbl 0812.57003)] for the quoted bound). Until advent of the quantum invariants, the best known lower bound on g(M) was the rank of the fundamental group of M, r( 1M). In [Invent. Math. 76, 455-468 (1984; Zbl 0538.57004)], Boileau and Zieschang presented a particular set of Seifert fibered spaces M, with the relatively rare property that g(M) = 3, while r( 1M) = 2. By studying the examples of Boileau and Zieschang, the author proves that quantum invariants may be used to provide a lower bound on g(M) which is both simpler to calculate and strictly larger than r( 1M).

2011 - Computational aspects of crystallization theory: complexity, catalogues and classifications of 3-manifolds [Articolo su rivista]
Bandieri, Paola; Casali, Maria Rita; Cristofori, Paola; Grasselli, Luigi; M., Mulazzani
abstract

The present paper is a survey of up-to-date results in 3-dimensional crystallization theory, in particular along the following directions:- generation and analysis of catalogues of PL-manifolds for increasing values of the vertex number of the representing graphs;- definition and/or computation of invariants for PL-manifolds, directly from the representing graphs.In particular, with regard to PL-manifold invariants, the authors focus on gems considered as an useful tool for computing Matveev complexity.

2011 - Gaifullin Alexander A., Configuration spaces, bistellar moves, and combinatorial formulae for the first Pontryagin class [Recensione] [Recensione in Rivista]
Casali, Maria Rita
abstract

The problem of finding explicit combinatorial formulae for the Pontryagin classes of triangulated manifolds was originally faced by Gabrielov, Gelfand and Losik in [Funct. Anal. Appl. 9, 103-115 (1975; Zbl 0312.57016)], and successively taken into account by various authors: see Russ. Math. Surv. 60, No. 4, 615-644 (2005; Zbl 1139.57026)] for a survey on the topic, with a comparison among different formulae. The present paper discusses the only two known combinatorial formulae for the first Pontryagin class that can be used for real computation, i.e. the classical Gabrielov-Gelfand-Losik formula and the local formula obtained by the author in [Izv. Math. 68, No. 5, 861-910 (2004; Zbl 1068.57022)]. Note that the first formula is presented not according to the original approach, based on endowing a triangulated manifold with locally flat connections, but according to MacPherson’s approach, based on the construction of a homology Gaussian mapping for a combinatorial manifold: see [Semin. Bourbaki, Vol. 1976/77, Lect. Notes Math. 677, 105-124 (1978; Zbl 0388.57013)]. A detailed exposition of the second formula is provided, together with the related notions and results concerning bistellar moves (see [Abh. Math. Semin. Univ. Hamb. 57, 69-86 (1987; Zbl 0651.52007)] and [Eur. J. Comb. 12, No.2, 129-145 (1991; Zbl 0729.52003)]) and the existence and uniqueness of universal local formulae for polynomials in rational Pontryagin classes. In the present paper the author succeeds in considerably simplifying his own explicit formula for the first Pontryagin class, by giving a new simpler algorithm for decomposing a cycle in the graph of bistellar moves of two-dimensional combinatorial spheres into a linear combination of elementary cycles.

2011 - REVIEW OF: "Howie James, Can Dehn surgery yield three connected summands?, Groups Geom. Dyn. 4, No. 4, 785-797 (2010)".[DE05880964X] [Recensione in Rivista]
Casali, Maria Rita
abstract

The {\it Cabling Conjecture} of Gonzales-Acuna and Short ([Math. Proc. Camb. Philos. Soc. 99, 89-102 (1986; Zbl 0591.57002)]) states that Dehn surgery on a knot in $\Bbb S^3$ can produce a reducible 3-manifold only if the knot is a cable knot and the surgery slope is that of a cabling annulus. Since the case of cable knot yields exactly two connected summands (see [Math. Proc. Camb. Philos. Soc. 102, No. 1, 97-101 (1987; Zbl 0655.57500)]), a weaker conjecture may be stated, asserting that a manifold obtained by Dehn surgery (i.e. $M^3=M(K, r)$, where $K$ is a knot in $\Bbb S^3$ and $r \in \Bbb Q \cup \{\infty\}$) cannot by expressed as a connected sum of three non-trivial manifolds. Note that results by [Topology Appl. 87, No.1, 73-78 (1998; Zbl 0926.57020)], [Topology Appl. 98, No.1-3, 355-370 (1999; Zbl 0935.57024)] and [J. Pure Appl. Algebra 173, No.2, 167-176 (2002; Zbl 1026.20019)] ensure that if $M(K, r)$ has three connected summands, than two of these must be lens spaces and the third must be a $\Bbb Z$-homology sphere. Moreover, in the same hypothesis, $r$ must be an integer: see [Math. Proc. Camb. Philos. Soc. 102, No. 1, 97-101 (1987; Zbl 0655.57500)]. \medskip The present paper faces the problem of three connected summands by making use of standard techniques of intersection graphs by Scharlemann and Gordon-Luecke. The main result proves that, if $K$ has bride-number $b$ and $M(K, r)= M_1 \#M_2\#M_3,$ $M_1$ and $M_2$ being lens spaces and $M_3$ being a homology sphere (but not a homotopy sphere), then $$|\pi_1(M_1)| + |\pi_1(M_2)| \le b+1.$$ As a consequence, the inequality $$|r| = |\pi_1(M_1)| \cdot |\pi_1(M_2)| \le \frac{b(b+2)}{4}$$ is obtained, which is a sharpening of a similar inequality due to Sayari (see [J. Knot Theory Ramifications 18, No. 4, 493-504 (2009; Zbl 1188.57004)]).

{\it Skein modules}, which are invariants of 3-manifolds as well as of links in these manifolds, were introduced by Przytycki ([Bull. Pol. Acad. Sci., Math. 39, No.1-2, 91-100 (1991; Zbl 0762.57013)]) and Turaev ([J. Sov. Math. 52, No.1, 2799-2805 (1990; Zbl 0706.57004)]). In the present paper, diagrams and Reidemeister moves for links in a twisted $\mathbb S^1$-bundles over a non-orientable surface are introduced, and the Kauffman bracket skein module (KBSM) of $\mathbb R P^3 \times \mathbb R P^3$ is computed. Note that the notion of diagrams of links in $F \times \mathbb S^1$ ($F$ being an orientable surface)), was introduced in [Topology Appl. 156, No. 10, 1831-1849 (2009; Zbl 1168.57010)], together with Reidemeister moves for such diagrams. After having extended the above notions to the case $N \hat \times \mathbb S^1$ ($N$ being a non-orientable surface), the author takes into account the particular case $N= \mathbb RP^2$, so that $N \hat \times \mathbb S^1 = \mathbb R P^3 \times \mathbb R P^3.$ The full computation of KBSM for $\mathbb R P^3 \times \mathbb R P^3$ shows that it has torsion (as it happens for $\mathbb S^1 \times \mathbb S^2$: see [Math. Z. 220, No.1, 65-73 (1995; Zbl 0826.57007)]), but - unlike the KBSM of $\mathbb S^1 \times \mathbb S^2$ - it does not split as a sum of cyclic modules. A new computation of KBSM of both $\mathbb S^1 \times \mathbb S^2$ and the lens space $L(p,1)$ completes the paper.

2011 - REVIEW OF: "Rafi Kasra - Schleimer Saul, Curve complexes are rigid, Duke Math. J. 158, No. 2, 225-246 (2011)".[DE059178142] [Recensione in Rivista]
Casali, Maria Rita
abstract

The {\it curve complex} of a surface was introduced into the study of Teichmüller space by Harvey (see [Riemann surfaces and related topics: Proc. 1978 Stony Brook Conf., Ann. Math. Stud. 97, 245-251 (1981; Zbl 0461.30036)]) as an analogous of the Tits building of a symmetric space. The present paper deals about geometric structure of the curve complex (see [Invent. Math. 138, No.1, 103-149 (1999; Zbl 0941.32012)] and [Geom. Funct. Anal. 10, No.4, 902-974 (2000; Zbl 0972.32011)]) of an orientable connected compact surface $S$, in case the {\it complexity} $\csi(S)$ of $S$ is greater or equal to two, where $\csi(S)=3g-3+b$, $g$ (resp. $b$) being the genus (resp. the number of boundary components) of $S$. By making use of the notions of {\it cobounded ending lamination} and of {\it marking complex}, together with some key results due to Gabai (see [Geom. Topol. 13, No. 2, 1017-1041 (2009; Zbl 1165.57015)]) and to Berhrstock-Kleiner-Minsky-Mosher (see [{\it Geometry and rigidity of mapping class group}, arXiv:0801.2006v4]), the authors prove that any quasi isometry of the curve complex is bounded distance from a simplicial automorphism. As a consequence, the quasi-isometry type of the curve complex determines the homeomorphism type of the surface.

2011 - REVIEW OF: "Spaggiari Fulvia, Regular genus and products of spheres, J. Korean Math. Soc. 47, No. 5, 925-934 (2010)".[DE057900743] [Recensione in Rivista]
Casali, Maria Rita
abstract

2011 - REVIEW OF: "Takao Kazuto, A refinement of Johnson's bounding for the stable genera of Heegaard splittings, Osaka J. Math. 48, No. 1, 251-268 (2011)".[DE058823344] [Recensione in Rivista]
Casali, Maria Rita
abstract

2010 - Characterization of minimal 3-manifolds by edge-coloured graphs [Articolo su rivista]
Casali, Maria Rita
abstract

We characterize combinatorial representations of minimal 3-manifolds by means of edge-coloured graphs. This enables their recognition among existing crystallization catalogues, and contemporarily enables the automatic construction of efficient and exhaustive catalogues representing all minimal 3-manifolds up to a fixed genus.

2010 - “Computational and Geometric Topology” - A conference in honour of Massimo Ferri and Carlo Gagliardi on their 60-th birthday. [Altro]
Bandieri, Paola; Casali, Maria Rita; A., Cattabriga; Cristofori, Paola; P., Frosini; Grasselli, Luigi; Landi, Claudia; M., Mulazzani
abstract

La conferenza ha inteso mettere in contatto ricercatori provenienti sia dall'ambito matematico che da quello ingegneristico, accomunati dall'interesse per tecniche topologiche di carattere geometrico e computazionale. Questi strumenti di ricerca sono essenziali in vari settori scientifici e per molteplici classi di applicazioni. In topologia geometrica risultano di particolare importanza le ricerche in teoria dei nodi, connesse allo studio di strutture biologiche (p.e. il confronto di dati genetici) e in fisica (con particolare riferimento alla teoria delle stringhe). La topologia computazionale si è invece rivelata indispensabile per la descrizione di forme al calcolatore e per la loro comparazione, con conseguenti ricadute nelle applicazioni che richiedono manipolazione grafica, confronto di modelli e reperimento di informazioni visuali. Tutto ciò ha ovvie importanti ricadute nel trattamento di dati in Internet. Tutti questi ambiti applicativi richiedono lo sviluppo di nuovi approcci teorici e competenze fortemente e intrinsecamente interdisciplinari, che l'iniziativa ha favorito.Il convegno si è articolato in sei conferenze su invito, tenute da alcuni tra i massimi esperti internazionali, della durata di 50 minuti ciascuna e da numerose comunicazioni di 30 minuti. Ha vauto lo scopo di divulgare nuovi risultati in Topologia Geometrica e Computazionale, ed ha coinvolto sia docenti che giovani ricercatori, nonché studenti di dottorato di ricerca in Matematica e/o in Ingegneria.Conferenzieri principali:Herbert Edelsbrunner (Duke University, Durham, NC, USA) Tomasz Kaczynski (Université de Sherbrooke, Canada)Sóstenes Lins (Departamento de Matemática, UFPE, Brasile)Sergei Matveev (Chelyabinsk State University, Russia) José María Montesinos (Universidad Complutense, Madrid, Spagna)Marian Mrozek (Jagiellonian University, Kraków, Polonia)

2010 - GEOMETRIA [Monografia/Trattato scientifico]
Casali, Maria Rita; Gagliardi, Carlo; Grasselli, Luigi
abstract

Il presente testo sviluppa argomenti tradizionalmente trattati nei corsi di “Geometria” (ovvero di “Algebra e Geometria”) nell'ambito delle lauree di primo livello, ed è particolarmente rivolto agli studenti delle Facoltà di Ingegneria e dei Corsi di Laurea in Matematica, Fisica ed Informatica. Il testo è suddiviso logicamente in due parti: - la prima parte contiene gli elementi fondamentali di Algebra lineare; - la seconda parte, di carattere più propriamente geometrico, riguarda le principali proprietà degli spazi euclidei, sviluppando in tale ambito la teoria delle coniche e delle quadriche. La presente edizione risulta integrata, rispetto a quella precedente, da una rivisitazione sostanziale della teoria delle coniche e delle quadriche, dall'inserimento di nuovi argomenti e complementi (algebre di Boole, isometrie del piano euclideo...), oltre che dalla aggiunta di nuove osservazioni ed esempi lungo tutto lo sviluppo del testo.L'esposizione risulta articolata, come ovvio per ogni teoria matematica, in Definizioni e Proposizioni (o Teoremi, nel caso in cui gli enunciati rivestano particolare importanza). Particolare rilievo viene attribuito ad Osservazioni ed Esempi atti a: - chiarire concetti, risultati, dimostrazioni; - stimolare i necessari collegamenti tra i vari argomenti; - motivare la genesi dei concetti e dei problemi; - evidenziare i casi notevoli di particolare rilievo nell'ambito di una teoria generale; - indicare possibili generalizzazioni o descrizioni alternative di una teoria. Ciò può consentire inoltre al Docente di “dosare” con maggiore libertà, secondo le proprie convinzioni ed esperienze didattiche, il peso da attribuire, durante le lezioni, ai vari argomenti del corso. Con l'eccezione delle principali proprietà degli insiemi numerici fondamentali e dell'utilizzo di una teoria “ingenua”, non rigorosamente assiomatica, degli insiemi (peraltro, brevemente richiamata nel primo Capitolo), il testo appare essenzialmente autocontenuto. In particolare, non risulta necessario alcun prerequisito di Geometria euclidea così come viene sviluppata, in modo sintetico, a partire da un sistema di assiomi, nelle Scuole secondarie. Seguendo l'impostazione algebrica ormai dominante nelle varie teorie matematiche e quindi in una ottica di “algebrizzazione della Geometria”, i concetti ed i risultati di natura geometrica, compresi quelli relativi alla Geometria euclidea, sono infatti ricavati da conoscenze di tipo algebrico precedentemente introdotte. Abbiamo cercato tuttavia di non fare perdere contenuto geometrico a tali concetti, sia mediante il metodo con cui questi vengono presentati, sia facendo spesso ricorso ad Osservazioni ed Esempi atti ad aiutare il lettore a ritrovare, pure in ambiti più generali, le proprietà geometriche già note. La scelta privilegiata è stata quella di sviluppare la teoria, sia dal punto di vista algebrico che da quello geometrico, per spazi di dimensione finita n; le dimensioni due e tre sono tuttavia sempre illustrate in modo dettagliato, come casi particolari e nelle loro specificità, sfruttandone le caratteristiche di rappresentatività. Tale scelta di generalità nella dimensione è dovuta essenzialmente a due considerazioni: da un lato riteniamo opportuno evitare inutili ripetizioni nella enunciazione della teoria per le varie dimensioni particolari, dall'altro siamo convinti che lo sviluppo della teoria in ambito ragionevolmente generale sia un ottimo stimolo allo sviluppo della capacità di astrazione e generalizzazione che è obiettivo fondamentale di ogni corso di matematica, anche nell'ambito dei nuovi ordinamenti degli studi universitari.

2010 - REVIEW OF: "Hongler Cam Van Quach - Weber Claude, Link projections and flypes, Acta Math. Vietnam. 33, No. 3, 433-457 (2008)". [DE055891064] [Recensione in Rivista]
Casali, Maria Rita
abstract

2010 - REVIEW of: "Todd Robert G., Khovanov homology and the twist number of alternating knots, J. Knot Theory Ramifications 18, No. 12, 1651-1662 (2009)". [DE056622768]. [Recensione in Rivista]
Casali, Maria Rita
abstract

2009 - REVIEW of: "Ayala R. - Fernández L.M. - Vilches J.A., Critical elements of proper discrete Morse functions, Mathematica Pannonica 19/2 (2008), 171-185" [DEB09154] [Recensione in Rivista]
Casali, Maria Rita
abstract

2009 - REVIEW of: "Cooper Daryl - Tillmann Stephan, The Thurston norm via normal surfaces,Pac. J. Math. 239, No. 1, 1-15 (2009)". [Zbl 1165.57018] [Recensione in Rivista]
Casali, Maria Rita
abstract

2009 - REVIEW of: "Dasbach Oliver T. - Lin, Xiao-Song, A volumish theorem for the Jones polynomial of alternating knots,Pac. J. Math. 231, No. 2, 279-291 (2007)". [Zbl 1166.57002] [Recensione in Rivista]
Casali, Maria Rita
abstract

2009 - REVIEW of: "Friedl S., Reidemeister torsion, the Thurston norm and Harvey's invariants, Pac. J. Math. 230, No. 2, 271-296 (2007)". [Zbl 1163.57008] [Recensione in Rivista]
Casali, Maria Rita
abstract

2009 - REVIEW of: "Thomassen Carsten - Vella Antoine, Graph-like continua, augmenting arcs, and Menger's theorem, Combinatorica 28, No. 5, 595-623 (2008)". [Zbl pre05580969] [Recensione in Rivista]
Casali, Maria Rita
abstract

2008 - A catalogue of orientable 3-manifolds triangulated by 30 coloured tetrahedra [Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola
abstract

The present paper follows the computational approach to 3-manifold classification via edge-coloured graphs, already performed in [1] (with respect to orientable 3-manifolds up to 28 coloured tetrahedra), in [2] (with respect to non-orientable3-manifolds up to 26 coloured tetrahedra), in [3] and [4] (with respect to genus two 3-manifolds up to 34 coloured tetrahedra): in fact, by automatic generation and analysis of suitable edge-coloured graphs, called crystallizations, we obtain a catalogue of all orientable 3-manifolds admitting coloured triangulations with 30 tetrahedra. These manifolds are unambiguously identified via JSJ decompositions and fibering structures. It is worth noting that, in the present work, a suitable use of elementary combinatorial moves yields an automatic partition of the elements of the generated crystallization catalogue into equivalence classes, which turn out to be in one-to one correspondence with the homeomorphism classes of the represented manifolds.

2008 - CRYSTALLIZATION CATALOGUES AND ARCHIVES OF CLOSED 3-MANIFOLDS WITH LOW GEM-COMPLEXITY [Software]
Casali, Maria Rita; Cristofori, Paola
abstract

CRYSTALLIZATION CATALOGUES is a collection of algorithmic procedures, which can be used to construct essential catalogues of bipartite and/or non bipartite edge-coloured graphs representing all orientable and/or non orientable 3-manifolds triangulated by a given number of coloured tetrahedra; the elements of the obtained catalogues may further be classified (i.e. subdivided into homeomorphism classes), as a first step toward the topological recognition of the involved manifolds. The output data of the C++ program (originally described in [M.R.Casali, Classification of non-orientable 3-manifolds admitting decompositions into 26 coloured tetrahedra, Acta Appl. Math. 54 (1999), 75-97]) generating catalogue C^2p of rigid bipartite crystallizations up to 2p vertices and/or catalogue ~C^2p of rigid non bipartite crystallizations up to 2p vertices are available, according to the number of vertices, at the Web page: http://cdm.unimo.it/home/matematica/casali.mariarita/CATALOGUES.htmThe Web page contains detailed results about existing catalogues ~C^26, C^28 and C^30 which are not included in the associated papers (for example: complete description of the involved manifolds, survey tables with related topological invariants, data about the reduced catalogues of cluster-less crystallizations…). Further, a comparative analysis of both complexity and geometric properties of manifolds represented by the subsequent subsets C^2p, p compreso tra 1 e 15, of all crystallizations in C^30 with exactly 2p vertices is also presented.

2008 - Gamma-class: A program to subdivide a set of rigid crystallizations of closed 3-manifolds into equivalence classes, whose elements represent homeomorphic manifolds [Software]
Casali, Maria Rita; Cristofori, Paola
abstract

Gamma-class is a program which implements the algorithm described in in [Casali M.R., Cristofori P., A catalogue of orientable 3-manifolds triangulated by 30 coloured tetrahedra, Journal of Knot Theory and its Ramifications 17 (2008), no.5, 579-599], with respect to a fixed (finite) set S of admissible sequences of elementary combinatorial moves: it yields, from any given list X of crystallizations, the automatic partition of the elements of X into equivalence classes, such that each class contains only crystallizations representing the same manifold. Moreover, the program tries the identification of the represented manifolds by means of comparison of the representatives of each class with known catalogues of crystallizations and/or splitting into connected sums.Program Gamma-class has already allowed the recognition and cataloguing of all manifolds represented by rigid bipartite and non bipartite crystallizations up to 30 vertices.

2008 - Representing and recognizing 3-manifolds obtained from I-bundles over the Klein bottle [Articolo su rivista]
Casali, Maria Rita
abstract

As it is well-known, the boundary of the orientable I-bundle $K X^sim I$ over the Klein bottle K is a torus; thus - in analogy with torus bundle construction (see [S]) - any integer matrix A of order two with determinant -1 (resp. +1) uniquely defines an orientable (resp. non-orientable) 3-manifold $(K X^sim I) cup (KX^sim I)/A$, which we denote by KB(A). In thepresent paper an algorithmic procedure is described, which allowsto construct, directly from any such matrix A, an edge-colouredgraph representing the manifold KB(A) associated to A. As a consequence, it is proved via regular genus (see [G]) thatthe Heegaard genus of any such manifold is less or equal to four;moreover, six elements of existing catalogues of orientable 3-manifolds represented by edge-coloured graphs (see [L] and [CC_2]) are directly recognized as manifolds of type KB(A).

2008 - REVIEW OF: "Burton Benjamin A., Enumeration of non-orientable 3-manifolds using face-pairing graphs and union-find, Discrete Comput. Geom. 38 (3), 527-571". [Zbl 1133.57001] [Recensione in Rivista]
Casali, Maria Rita
abstract

2008 - REVIEW OF: "Chen Yichao - Liu Yanpei, A note on lower bounds for maximum genus, Util. Math. 73 (2007), 23-31".[Zbl 1138.05016] [Recensione in Rivista]
Casali, Maria Rita
abstract

2008 - REVIEW OF: "Falbel Elisha, A spherical CR structure on the complement of the figure eight knot with discrete holonomy, J. Differ. Geom. 79 (1) (2008), 69-110".[Zbl 1148.57025] [Recensione in Rivista]
Casali, Maria Rita
abstract

2008 - REVIEW OF: "Flapan Erica - Naimi Ramin, The Y-triangle move does not preserve intrinsic knottedness, Osaka J. Math. 45 (1) (2008), 107-111".[Zbl 1145.05019] [Recensione in Rivista]
Casali, Maria Rita
abstract

2008 - REVIEW OF: "Funar Louis, Surface cubications mod flips, Manuscr. Math. 125 (3), 285-307".[Zbl 1144.05022] [Recensione in Rivista]
Casali, Maria Rita
abstract

2008 - REVIEW OF: "Li Tao, Saddle tangencies and the distance of Heegaard splittings, Algebr. Geom. Topol. 7 (2007), 1119-1134". [Zbl 1134.57005] [Recensione in Rivista]
Casali, Maria Rita
abstract

2008 - REVIEW OF: "Poenaru V. - Tanasi C., Some remarks on geometric simple connectivity in dimension four - part A, Rend. Semin. Mat. Torino 65 (3) (2007), 313-344". [Zbl 1150.57005] [Recensione in Rivista]
Casali, Maria Rita
abstract

2008 - REVIEW OF: "Scharlemann Martin, Generalized property R and the Schoenflies conjecture, Comment. Math. Helv. 83 (1) (2008), 421-449". [Zbl 1148.57032] [Recensione in Rivista]
Casali, Maria Rita
abstract

2008 - REVIEW OF: "Taylor Scott A., On non-compact Heegaard splittings, Algebr. Geom. Topol. 7 (2007), 603-672". [Zbl 1134.57006] [Recensione in Rivista]
Casali, Maria Rita
abstract

2008 - REWIEW OF: "Przytycki, Piotr, A The fixed point theorem for simplicial nonpositive curvature, Math. Proc. Camb. Philos. Soc. 144(3) (2008), 683-695"[Zbl 1152.20038] [Recensione in Rivista]
Casali, Maria Rita
abstract

2007 - DUKE III: A program to handle edge-coloured graphs representing PL n-dimensional manifolds [Software]
Casali, Maria Rita; Cristofori, Paola
abstract

One of the main features of crystallization theory relies on the purely combinatorial nature of the representing objects, which makes them particularly suitable for computer manipulation. This fact allows a computational approach to the study of PL n-manifolds, which has been performed by means of several functions, collected in a unified program, called DUKE III. DUKE III allows automatic manipulation of edge-coloured graphs representing PL n-manifolds (code computation, checking possible isomorphism between edge-coloured graphs, construction of boundary graph, checking bipartition, connectedness, rigidity and planarity conditions, combinatorial moves, invariants computation...). Furthermore, DUKE III allows automatic recognition of orientable 3-manifolds triangulated by at most 30 coloured tetrahedra and of non-orientable 3-manifolds triangulated by at most 26 coloured tetrahedra (by making use of existing electronic archives of all rigid bipartite crystallizations up to 30 vertices and non-bipartite ones up to 26 vertices, due to the same research team).

2007 - Estimating Matveev's complexity via crystallization theory [Articolo su rivista]
Casali, M. R.
abstract

In [M.R. Casali, Computing Matveev's complexity of non-orientable 3-manifolds via crystallization theory, Topology Appl. 144(1-3) (2004) 201-209], a graph-theoretical approach to Matveev's complexity computation is introduced, yielding the complete classification of closed non-orientable 3-manifolds up to complexity six. The present paper follows the same point-of view, making use of crystallization theory and related results (see [M. Ferri, Crystallisations of 2-fold branched coverings of S^3, Proc. Amer. Math. Soc. 73 (1979) 271-276]; [M.R. Casali, Coloured knots and coloured graphs representing 3-fold simple coverings of S^3, Discrete Math. 137 (1995) 87-98]; [M.R. Casali, From framed links to crystallizations of bounded 4-manifolds, J. Knot Theory Ramifications 9(4) (2000) 443-458]) in order to significantly improve existing estimations for complexity of both 2-fold and three-fold simple branched coverings (see [O.M. Davydov, The complexity of 2-fold branched coverings of a 3-sphere, Acta Appl. Math. 75 (2003) 51-54] and [O.M. Davydov, Estimating complexity of 3-manifolds as of branched coverings, talk-abstract, Second Russian-German Geometry Meeting dedicated to 90-anniversary of A.D.Alexandrov, Saint-Petersburg, Russia, June 2002]) and 3-manifolds seen as Dehn surgery (see [G. Amendola, An algorithm producing a standard spine of a 3-manifold presented by surgery along a link, Rend. Circ. Mat. Palermo 51 (2002) 179-198]).

2007 - TORUS BUNDLE: A program to construct edge-coloured graphs representing torus bundles over the circle. [Software]
Casali, Maria Rita
abstract

TORUS BUNDLE is based on an algorithmic construction of edge-coloured graphs representing 3-manifolds, which are torus bundles over S^1 (see [M.R.Casali, Representing and recognizing torus bundles over S^1, Boletín de la Sociedad Matemática Mexicana, 10(3) (2005), 89-106]). The algorithm starts from a regular integer matrix A which describes the monodromy and contains at least one zero element. This program has allowed the topological recognition of all torus bundles among the 3-manifolds represented by the existing crystallization catalogues C^28 and ~C^26. Note that TORUS BUNDLE is actually independent from DUKE III, but related to it: it supplies - as output - the code of an edge-coloured graph which can be successively inserted in DUKE III in order to be analyzed and simplified, for recognition of the represented manifold....

2006 - c_GM: A program to compute GM-complexity of edge-coloured graphs representing closed 3-manifolds [Software]
Casali, Maria Rita; Cristofori, Paola
abstract

c_GM is a C++ program which implements the algorithmic procedure described in [M.R. Casali, Computing Matveev's complexity of non-orientable 3-manifolds via crystallization theory, Topology and its Applications 144 (1-3) (2004), 201-209], to estimate Matveev's complexity of a 3-manifold starting from the code of an associated edge-coloured graph (GM-complexity computation). This program has already allowed to compute GM-complexity of all non-orientable 3-manifolds represented by edge-coloured graphs up to 26 vertices (catalogue ~C26) and of all orientable 3-manifolds represented by edge-coloured graphs up to 28 vertices (catalogue C28), giving a significant help to the classification of the involved manifolds; classes of manifolds for which the estimation is actually exact have been also detected. Furthermore, a comparison between different notions of complexity has been performed with the aid of this program: see [M.R. Casali, Computing Matveev's complexity of non-orientable 3-manifolds via crystallization theory, Topology and its Applications 144 (1-3) (2004), 201-209] and [M.R. Casali - P.Cristofori, Computing Matveev's complexity via crystallization theory: the orientable case, Acta Applicandae Mathematicae 92 (2006), 113-123]. The program computes the GM-complexity both of a single edge-coloured graph and of a list of edge-coloured graphs. It also computes the minimal GM-complexity of a set of crystallizations representing the same manifold, thus providing upper bounds for the complexity of the manifold itself.c_GM interacts with Duke III program for handling edge-coloured graphs, since it recognizes Duke’s encoding of graphs and it can run on catalogues of crystallizations generated and classified through the procedures of CRYSTALLIZATION CATALOGUES and program Gamma_class.

2006 - Computing Matveev's complexity via crystallization theory: the orientable case [Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola
abstract

By means of a slight modification of the notion of GM-complexity introduced in [Casali, M.R., Topol. Its Appl., 144: 201-209, 2004], the present paper performs a graph-theoretical approach to the computation of (Matveev's) complexity for closed orientable 3-manifolds. In particular, the existing crystallization catalogue C-28 available in [Lins, S., Knots and Everything 5, World Scientific, Singapore, 1995] is used to obtain upper bounds for the complexity of closed orientable 3-manifolds triangulated by at most 28 tetrahedra. The experimental results actually coincide with the exact values of complexity, for all but three elements. Moreover, in the case of at most 26 tetrahedra, the exact value of the complexity is shown to be always directly computable via crystallization theory.

2004 - Computing Matveev’s complexity of non-orientable 3-manifolds via crystallization theory [Articolo su rivista]
Casali, Maria Rita
abstract

The present paper looks at Matveev's complexity (introduced in 1990 and based on the existence of a simple spine for each compact 3-manifold: see [Acta Appl. Math. 19 (1990), 101-130]) through another combinatorial theory for representing 3-manifolds, which makes use of particular edge-coloured graphs, called crystallizations. Crystallization catalogue $\tilde C^{26}$ for closed non-orientable 3-manifolds (due to [Acta Appl. Math. 54 (1999), 75-97]) is proved to yield upper bounds for Matveev's complexity of any involved 3-manifold. As a consequence, an improvement of Amendola and Martelli classification of closed non-orientable irreducible and $P^2$-irreducible 3-manifolds up to complexity c=6 is obtained.

2004 - Dotted links, Heegaard diagrams and coloured graphs for PL 4-manifolds [Articolo su rivista]
Casali, Maria Rita
abstract

The present paper is devoted to establish a connection between the 4-manifold representation method by dotted framed links (or - in the closed case - by Heegaard diagrams) and the so called crystallization theory, which visualizes general PL-manifolds by means of edge-coloured graphs.In particular, it is possible to obtain a crystallization of aclosed 4-manifold $M^4$ starting from a Heegaard diagram$(\#_m(S^1 X S^2), \omega),$ and the algorithmicity of the whole process depends on the effective possibility of recognizing $(\#_m(S^1 X S^2), \omega)$ to be a Heegaard diagram by crystallization theory.

2004 - Representing and recognizing torus bundles over S1 [Articolo su rivista]
Casali, Maria Rita
abstract

As it is well-known, torus bundles over $S^1$ areidentified by means of regular integer matrices of order two (see[S]); in the present paper an algorithmic procedure is described,which allows to construct, directly from any matrix $A \in GL (2;Z)$, an edge-coloured graph representing the torus bundle$TB(A)$ associated to $A.$ As a consequence, five topologically undetected elements of Lins's catalogue of orientable 3-manifolds (see [L]) are finally recognized as torus bundles over $S^1$.

2003 - On the regular genus of 5-manifolds with free fundamental group [Articolo su rivista]
Casali, Maria Rita
abstract

In the present paper, we obtain the following classification of closed orientable PL 5-manifolds M^5 with free fundamental group of rank m, so that the difference between the regular genus G(M^5) and m is less or equal to eight: (a) G(M^5) = m iff $M^5= \#_m (S^1 X S^4)$; (b) it is impossible m+1 less than or equal to G(M^5) less than or equal to m+7; (c) if G(M^5) = m+8, then either $M^5= \#_m (S^1 X S^4) # (S^2 X S^3)$ or $M^5= \#_m (S^1 X S^4) # (S^2 X_\sim S^3)$. As a consequence, we complete the classification of PL 5-manifolds up to regular genus eight, and compute the regular genus of the 5-dimensional real projective space RP5: if G(M^5) = 8, then either $M^5= S^2 X S^3$ or $M^5= S^2 X_\sim S^3$ or $M^5= \#_8 (S^1 X S^4)$; $G(S^2 X S^3) = 8$; $G(S^2 X_\sim S^3)$ greater than or equal to 8; $G(RP^5) = 9$.

2002 - An equivalence criterion for PL-manifolds [Articolo su rivista]
Casali, Maria Rita
abstract

Within geometric topology of PL n-manifolds (with or withoutboundary), a representation theory exists, which makes use of (n+1)-coloured graphs. Aim of this paper is to translate the homeomorphism problem for the represented manifolds into an equivalence problem for (n+1)-coloured graphs, by means of a finite number of graph-moves, called dipole moves. Actually,the same problem was already faced - and solved - in [FG], but for closed n-manifolds, only; here, the whole class of PL-manifolds is considered, and the up-to-date knowledge involved in the proof (i.e. shelling and bistellar operations, among all) throw a deeper light on the previous results, too. Moreover, the equivalence criterion for PL-manifolds via dipole moves is proved to be equivariant with respect to the boundary triangulation.

2001 - A code for m-bipartite edge-coloured graphs [Articolo su rivista]
Casali, Maria Rita; Gagliardi, Carlo
abstract

An (n+1)-coloured graph $(\Gamma,\gamma)$ is said to be m-bipartite if m is the maximum integer so that every m-residue of $(\Gamma,\gamma)$ (i.e. every connected subgraph whose edges are coloured by only m colours) is bipartite; obviously, every (n+1)-coloured graph, with $n \ge 2$, results to be m-bipartite for some m, with $\ 2 \le m \le n+1$. In this paper, a numerical code of length $(2n-m+1) \times q$ is assigned to each m-bipartite (n+1)-coloured graph of order 2q.Then, it is proved that any two such graphs have the same code if and only if they are colour-isomorphic, i.e. if a graph isomorphism exists, which transforms the graphs one into the other, up to permutation of the edge-colouring. More precisely, if H is a given group of permutations on the colour set, we face the problem of algorithmically recognizing H-isomorphic coloured graphs by means of a suitable definition of H-code.

2001 - Geometria [Monografia/Trattato scientifico]
Casali, Maria Rita; Grasselli, Luigi; Gagliardi, Carlo
abstract

Il presente testo sviluppa argomenti tradizionalmente trattati nei corsi di “Geometria” (ovvero di “Algebra e Geometria”) nell'ambito delle lauree di primo livello, ed è particolarmente rivolto agli studenti delle Facoltà di Ingegneria e dei Corsi di Laurea in Matematica, Fisica ed Informatica.Il testo è suddiviso logicamente in due parti:- la prima parte contiene gli elementi fondamentali di Algebra lineare;- la seconda parte, di carattere più propriamente geometrico, riguarda le principali proprietà degli spazi euclidei, sviluppando in tale ambito la teoria delle coniche e delle quadriche.L'esposizione risulta articolata, come ovvio per ogni teoria matematica, in Definizioni e Proposizioni (o Teoremi, nel caso in cui gli enunciati rivestano particolare importanza). Particolare rilievo viene attribuito ad Osservazioni ed Esempi atti a:- chiarire concetti, risultati, dimostrazioni;- stimolare i necessari collegamenti tra i vari argomenti;- motivare la genesi dei concetti e dei problemi;- evidenziare i casi notevoli di particolare rilievo nell'ambito di una teoria generale;- indicare possibili generalizzazioni o descrizioni alternative di una teoria.Ciò può consentire inoltre al Docente di “dosare” con maggiore libertà, secondo le proprie convinzioni ed esperienze didattiche, il peso da attribuire, durante le lezioni, ai vari argomenti del corso.Con l'eccezione delle principali proprietà degli insiemi numerici fondamentali e dell'utilizzo di una teoria “ingenua”, non rigorosamente assiomatica, degli insiemi (peraltro, brevemente richiamata nel primo Capitolo), il testo appare essenzialmente autocontenuto. In particolare, non risulta necessario alcun prerequisito di Geometria euclidea così come viene sviluppata, in modo sintetico, a partire da un sistema di assiomi, nelle Scuole secondarie.Seguendo l'impostazione algebrica ormai dominante nelle varie teorie matematiche e quindi in una ottica di “algebrizzazione della Geometria”, i concetti ed i risultati di natura geometrica, compresi quelli relativi alla Geometria euclidea, sono infatti ricavati da conoscenze di tipo algebrico precedentemente introdotte. Abbiamo cercato tuttavia di non fare perdere contenuto geometrico a tali concetti, sia mediante il metodo con cui questi vengono presentati, sia facendo spesso ricorso ad Osservazioni ed Esempi atti ad aiutare il lettore a ritrovare, pure in ambiti più generali, le proprietà geometriche già note.La scelta privilegiata è stata quella di sviluppare la teoria, sia dal punto di vista algebrico che da quello geometrico, per spazi di dimensione finita n; le dimensioni due e tre sono tuttavia sempre illustrate in modo dettagliato, come casi particolari e nelle loro specificità, sfruttandone le caratteristiche di rappresentatività. Tale scelta di generalità nella dimensione è dovuta essenzialmente a due considerazioni: da un lato riteniamo opportuno evitare inutili ripetizioni nella enunciazione della teoria per le varie dimensioni particolari, dall'altro siamo convinti che lo sviluppo della teoria in ambito ragionevolmente generale sia un ottimo stimolo allo sviluppo della capacità di astrazione e generalizzazione che è obiettivo fondamentale di ogni corso di matematica, anche nell'ambito dei nuovi ordinamenti degli studi universitari.

2001 - Representing manifolds by crystallization theory: foundations, improvements and related results [Articolo su rivista]
Bandieri, Paola; Casali, Maria Rita; Gagliardi, Carlo
abstract

Crystallization theory was born in Italy during the 70's, due to Mario Pezzana and his school, as a combinatorial representation tool for piecewise-linear (PL) manifolds of arbitrary dimension. The present paper performes a - not exaustive - survey of the main results of PL-topology achieved through crystallization theory, both by the italian school of M. Pezzana and by other researchers of different schools, which contributed to the development of the ideas. In many cases, research problems and open questions are reviewed, together with the most recent - sometimes unpublished - results.

2000 - From framed links to crystallizations of bounded 4-manifolds [Articolo su rivista]
Casali, Maria Rita
abstract

It is well-known that every 3-manifold $M^3$ may be represented by a framed link (L, c), which indicates the Dehn-surgery from $S^3$ to $M^3 = M^3(L, c)$; moreover, $M^3$ is the boundary of a PL 4-manifold $M^4 = M^4(L, c)$, which is obtained from $D^4$ by adding 2-handles along the framed link (L, c). In this paper we study the relationships between the above representations and the representation theory of general PL-manifolds by edge-coloured graphs: in particular, we describe how to construct a 5-coloured graph representing $M^4 = M^4(L, c)$, directly from a planar diagram of (L, c). As a consequence, relations between the combinatorial properties of the link L and both the Heegaard genus of $M^3 = M^3(L, c)$ and the regular genus of $M^4 = M^4(L, c)$ are obtained.

1998 - Average order of coloured triangulations: The general case [Articolo su rivista]
Casali, Maria Rita
abstract

In [Combinatorics of triangulations of 3-manifolds, Trans. Amer. Math. Soc. 337 (2) (1993), 891-906], Luo and Stong introduced the notion of "average edge order" $\mu_0(K) = \frac {3 F_0(K)}{E_0(K),$ K being a triangulation of a closed 3-manifold M with $E_0(K)$ edges and $F_0(K)$ triangles. The present paper extends the above notion to the "average (n-2)-simplex order" of a coloured triangulation K of a compact PL n-manifold $M^n$ with $\alpha_i(K)$ i-simplices: $\mu(K) = \frac {n \alpha_{n-1}(K)}{\alpha_{n-2}(K)$.Main properties of $\mu(K)$ and its relations with the topology of $M^n$, both in the closed and bounded case, are investigated; the obtained results show the existence of strong analogies with the 3-dimensional simplicial case (see the quoted paper by Luo and Stong, together with [The average edge order of triangulations of 3-manifolds, Osaka J. Math. 33(1986), 761-773] by Tamura).

1998 - Classification of nonorientable 3-manifolds admitting decompositions into <= 26 coloured tetrahedra [Articolo su rivista]
Casali, Maria Rita
abstract

The present paper adopts a computational approach to the study of nonorientable 3-manifolds: in fact, we describe how to create an automatic catalogue of all nonorientable 3-manifolds admitting coloured triangulations with a fixed number of tetrahedra. In particular, the catalogue has been effectively produced and analysed for up to 26 tetrahedra, to reach the complete classification of all involved 3-manifolds. As a consequence, the following summarising result can be stated: THEOREM I. Exactly seven closed connected prime nonorientable 3-manifolds exist, which admit a coloured triangulation consisting of at most 26 tetrahedra. More precisely, they are the four Euclidean nonorientable 3-manifolds, the nontrivial $S^2$-bundle over $S^1$, the topological product between the real projective plane $RP^2$ and $S^1$, and the torus bundle over $S^1$, with monodromy induced by matrix (0,1;1,-1).

1997 - A combinatorial proof of Rohlin Theorem [Articolo su rivista]
Casali, Maria Rita; Gagliardi, Carlo
abstract

We present an algorithmic and combinatorial proof of the following well-known theorem, originally proved by Rohlin: Every closed orientable 3-manifold $M^3$ bounds a simply connected orientable 4-manifold $M^4$.' More precisely, an edge-coloured graph representing $M^4$ is obtained as the final result of a finite and well-determined sequence of admissible moves', starting from any given edge-coloured graph representing $M^3$.

1997 - An equivalence criterion for 3-manifolds [Articolo su rivista]
Casali, Maria Rita
abstract

Within geometric topology of 3-manifolds (with or withoutboundary), a representation theory exists, which makes use of 4-coloured graphs. Aim of this paper is to translate the homeomorphism problem for the represented manifolds into an equivalence problem for 4-coloured graphs, by means of a finite number of graph-moves, called "dipole moves". Moreover, interesting consequences are obtained, which are related with the same problem in the n-dimensional setting.

1997 - Classifying PL 5-manifolds by regular genus: the boundary case [Articolo su rivista]
Casali, Maria Rita
abstract

In the present paper we face the problem of classifying classes of orientable PL 5-manifolds $M^5$ with $h\ge1$ boundary components, by making use of a combinatorial invariant called “regular genus” $G(M^5)$. In particular, a complete classification up to regular genus five is obtained: if $G(M^5)=\rho \le 5,$ then $M^5= \#_{\rho -\rho’} (S^1 x S^4) \# H_{\rho’}^h$where $\rho’=G(\partial M^5}$ denotes the regular genus of the boundary $\partial M^5$ and $H_{\rho’}^h$ denotes the connected sum of h orientable 5-dimensional handlebodies $Y_{\alpha_i}$ of genus $\alpha_i \ge 0$ so that $\sum_{i=1,…,h} \alpha_i = \rho’$. Moreover, we give a characterization of orientable PL 5-manifolds $M^5$ with boundary satisfying particular conditions related to the “gap” between $G(M^5)$ and either $G(\partial M^5}$ or the rank of their fundamental group. Further, the paper explains how the above results (together with other known properties of regular genus of PL-manifolds) may lead to a combinatorial approach to 3-dimensional Poincarè Conjecture.

1997 - Geometric topology by crystallization theory: results and problems [Articolo su rivista]
Casali, Maria Rita
abstract

Within geometric (or PL) topology, a representation theory exists, which makes use of a particular class of edge-coloured graphs - called crystallizations - to deal with PL-manifolds of arbitrary dimension, with or without boundary. The present paper is mainly devoted to review some recent developments of crystallization theory, and to show the existingrelationships with other "classical" representation methods for PL-manifolds, such as Heegaard splittings, branched coverings and surgery on framed links.

1997 - Handle-decompositions of PL 4-manifolds [Articolo su rivista]
Casali, Maria Rita; L., Malagoli
abstract

The present paper studies the relationship between handle-decompositions of PL 4-manifolds and the so called "crystallization theory", which represents PL n-manifolds by means of pseudosimplicial triangulations admitting exactly n+1 vertices. Within this theory, the combinatorial invariant "regular genus" plays a central role. In particular, the authors obtain the characterization of PL 4-manifolds $M^4$ (with empty or connected boundary $\partial M^4$) satisfying suitable conditions concerning the regular genus $G(M^4)$, the rank of the fundamental group $rk(\pi_1(M^4))$, and the boundary regular genus $G(\partial M^4)$. As a consequence, it is completed the classification of PL 4-manifolds with (possibly disconnected) boundary up to regular genus two.

1996 - An infinite class of bounded 4-manifolds having regular genus three [Articolo su rivista]
Casali, Maria Rita
abstract

In the present paper the classification of PL 4-manifolds by means of the combinatorial invariant “regular genus” is proved to be not finite to one: indeed, the set of all $D^2$-bundles over $S^2$ (i.e. every bundle $\csi_c$ with Euler class $c$ and boundary L(c,1), $c \in Z-\{0,-1,-1}$, together with the trivial bundle $S^2 X D^2$) constitutes an infinite family of PL 4-manifolds with the same regular genus (equal to three). Further, general results are obtained, concerning PL 4-manifolds with “restricted gap” between their regular genus and the rank of their fundamental group, especially in case of free fundamental group.

1996 - Equivalenze combinatorie di varietà bidimensionali [Articolo su rivista]
Casali, Maria Rita; R., Panisi
abstract

In this work, a set of combinatorial moves is determined, which realize the homeomorphism of bidimensional manifolds, working on their coloured triangulations, or –equivalently – on the edge-coloured graphs representing them. In particular, the obtained moves result to be equivariant with respect to the manifold boundary.

1996 - Sul genere regolare delle PL-varietà con gruppo fondamentale libero. [Abstract in Atti di Convegno]
Casali, Maria Rita
abstract

All'interno della teoria di rappresentazione delle n-varietà PL tramite grafi colorati sugli spigoli ([M.Pezzana, Sulla struttura topologica delle varietà compatte, Atti Sem. Mat. Fis. Univ. Modena, 23 (1974), 269-277], [M.Ferri - C.Gagliardi - L.Grasselli, A graph-theoretical representation of PL-manifolds. A survey on crystallizations, Aequationes Mat. 31, (1986), 121-141], [A.Vince, n-graphs, Discrete Math. 72 (1988), 367-380], [A.Costa, Coloured graphs representing manifolds and universal maps, Geom. Dedicata 28 (1988), 349-357], [S.Lins, Gems, computers and attractors for 3-manifolds, Knots and Everything, World Scientific 5, 1995]…), è nota l'esistenza di un invariante per varietà PL - detto genere regolare - che estende a dimensione arbitraria le classiche nozioni di genere di una superficie e di genere di Heegaard di una 3-varietà (si veda [C.Gagliardi, Extending the concept of genus to dimension n, Proc. Amer. Math. Soc. 81 (1981), 473-481]). In qualunque dimensione, sia nel caso chiuso che nel caso con bordo, è facile verificare che $G(M^n) \ge rk(M^n)$, ove $G(M^n)$ denota il genere regolare della n-varietà $M^n$ e $rk(M^n)$ denota il rango del suo gruppo fondamentale $\pi_1(M^n)$. Nella presente comuncazione sono esposti alcuni risultati ottenuti in dimensione quattro e cinque sulla classificazione delle PL-varietà con gruppo fondamentale libero, quando sia nota la differenza tra il genere regolare della varietà e il rango del suo gruppo fondamentale. Di seguito è esposto il primo di tali risultati, che riguarda il caso in cui la differenza descritta sia nulla.Sia $M^n$ una n-varietà PL di dimensione n, con n=4 o n=5. Allora: $G(M^n) = m$ e $rk(M^n) se e solo se$M^n = #m(S^1 x S^{n-1}$(se$\partial M^n =\emptyset$) o$M^n = #l(S^1 x S^{n-1} # Y^n_{m-l}$(se$\partial M^n \ne \emptyset$), ove$#a(S^1 x S^{n-1}$denota la somma connessa di a copie del fibrato standard su$S^1$con fibra$S^{n-1}$, mentre$Y^n_a$denota il corpo di manici n-dimensionale di genere a. Come conseguenza dei risultati ottenuti, si caratterizzano le varietà$M^n$con$G(M^n) \le G(\partial M^n) +1$(per n=4 e per n=5), e si completano le classificazioni delle 4-varietà con bordo fino a genere regolare tre, delle 5-varietà con bordo fino a genere regolare cinque e delle 5-varietà chiuse fino a genere regolare otto. 1995 - A note about bistellar operations on PL-manifolds with boundary [Articolo su rivista] Casali, Maria Rita abstract In 1990, U. Pachner proved that simplicial triangulations of the same PL-manifold (with boundary) are always connected by a finite sequence of transformations belonging to two different groups: shelling operations (and their inverses), which work mostly with the boundary triangulations, and bistellar operations, which affect only the interior of the triangulations. The purpose of this note is to prove that, in case of simplicial triangulations coinciding on the boundary, bistellar operations are sufficient to solve the homeomorphism problem. 1995 - A note about the closing S3 recognition algorithm [Articolo su rivista] Casali, Maria Rita abstract In [V2], Vince outlined three potential graph algorithms for$S^3$recognition, namely shelling, reducing, and closing; on the other hand, he conjectured that the graph$H_0$of Fig.1 - which proves the first two to fail - could be a counterexample for the third one, too. This note shows that the conjecture is false; so, the validity of the closing algorithm is still an open problem. 1995 - Coloured knots and coloured graphs representing 3-fold simple coverings of S3 [Articolo su rivista] Casali, Maria Rita abstract It is well-known that every closed orientable 3-manifold$M^3$is the 3-fold simple covering$M^3(K,\omega)$of$S^3$branched over a knot K: hence,$M^3$may be visualized by the associated coloured knot$(K,\omega)$. On the other hand, PL-manifolds of arbitrary dimension may be represented by coloured graphs, via pseudosimplicial triangulations. The present paper produces an algorithm to construct a 4-coloured graph representing$M^3(K,\omega)$, directly 'drawn over' the coloured knot$(K,\omega)$. 1994 - Classifying PL 5-manifolds up to regular genus seven [Articolo su rivista] Casali, Maria Rita; Gagliardi, Carlo abstract In the present paper we show that the only closed orientable PL 5-manifolds of regular genus less or equal to seven are the 5-sphere$S^5$and the connected sum of m copies of$S^1 X S^4$, with$m \le 7$. As a consequence, the genus of$S^3 X S^2$is proved to be eight. This suggests a possible approach to the (3-dimensional) Poincarè Conjecture, via the well-known classification of simply connected 5-manifolds, obtained by Smale and Barden. 1994 - The average edge order of 3-manifold coloured triangulations [Articolo su rivista] Casali, Maria Rita abstract If K is a triangulation of a closed 3-manifold M with$E_0(K)$edges and$F_0(K)$triangles, then the average edge order of K is defined to be$\mu_0(K) = 3F_0(K)/E_0(K) In [8], the relations between this quantity and the topology of M are investigated, especially in the case of $\mu_0(K)$ being small (where the study relies on Oda's classification of triangulations of $S^2$ up to eight vertices. In the present paper, the attention is fixed upon the average edge order of coloured triangulations; surprisingly enough, the obtained results are perfectly analogous to Luo-Stong' ones, and may be proved with little effort by means of edge-coloured graphs representing manifolds.

1993 - A note on the characterization of handlebodies [Articolo su rivista]
Casali, Maria Rita
abstract

The work is devoted to extend to dimension five the following combinatorial characterization of (orientable and non-orientable) handlebodies, already proved for dimensions three and four by the same author: a compact connected 5-manifold $M^5$ is a handlebody (of genus g) iff $G(M^5)= G(\partial M^5} (=g)$, $G(X) being the regular genus of the manifold X. Moreover, partial results in dimension n induce to conjecture that an analogous characterization also holds for handlebodies of arbitrary dimension. 1993 - A universal branching set for 4-dimensional manifolds [Articolo su rivista] Casali, Maria Rita abstract In this work, a universal branching set K for orientable 4-manifolds, such that$\pi_1(S^4 - K) = [a, b, c/aca^{-1}c^{-1} =1]$is proved to exist. This leads to the possibility of representing every closed connected orientable 4-manifold by a suitable transitive set$\{\sigma, \tau, \mu\}$of permutations, in analogy with known results for dimension three (see [Montesinos] and [Costa-delValMelus]). 1992 - A combinatorial characterization of 4-dimensional handlebodies [Articolo su rivista] Casali, Maria Rita abstract In this work, (orientable and non-orientable) 4-dimensional handlebodies are proved to be the only 4-manifolds whose regular genus equals the one of their boundary. As a consequence, we obtain the classification of all 4-manifolds of regular genus g less-than-or-equal-to 1. 1992 - Two-fold branched coverings of S3 have type six [Articolo su rivista] Casali, Maria Rita abstract In this work we prove that every closed, orientable 3-manifold$M^3$which is a two-fold covering of$S^3$branched over a link, has type six. This implies that$M^3$is the quotient of the universal pseudocomplex K(4,6) by the action of a finite index subgroup of a fuchsian group with presentation S(4,6)= < a_1, a_2, a_3, a_4 / (a_1)^3 = (a_2)^3 = (a_3)^3 = (a_4)^3 = a_1 a_2 a_3 a_4 =1 >Moreover, the same result is proved to be true in case$M^3$being an unbranched covering of a two-fold branched covering of$S^3$. 1991 - CLASSIFICAZIONE DELLE 5-VARIETA' P.L. CON GENERE REGOLARE <= 8 [Abstract in Atti di Convegno] Casali, Maria Rita abstract Il genere regolare di una n-varietà PL$M^n$è un invariante combinatorio$G(M^n) \ge 0$, definito in C.Gagliardi, Extending the concept of genus to dimension n, Proc. Amer. Math. Soc. 81 (1981), 473-481]) all'interno della teoria di rappresentazione delle varietà tramite grafi colorati sugli spigoli, che estende a dimensione arbitraria le classiche nozioni di genere di una superficie e di genere di Heegaard di una 3-varietà. E' noto che, in ogni dimensione,$G(M^n)$assume valore zero se e soltanto se$M^n$è omeomorfa alla n-sfera$S^n.$Per quanto riguarda la dimensione cinque, si è ottenuta la classificazione completa delle varietà chiuse connesse ed orientabili aventi genere regolare minore o uguale ad otto: Sia$M^5$una 5-varietà PL chiusa connessa orientabile. Allora: (a)$1 \le G(M^n) = m \le 7$se e solo se$M^5 = #m(S^1 x S^4);$(b)$G(M^n) = 8$se e solo se o$M^5 = #8(S^1 x S^4)$o$M^5 = S^2 x S^3$(ove$#m(S^1 x S^4)$denota la somma connessa di m copie di$S^1 x S^4$). Come conseguenza, si è calcolato il genere regolare dello spazio proiettivo reale$RP^5$:$G(RP^5) =9.$1991 - 2-symmetric crystallizations and 2-fold branched coverings of S3 [Articolo su rivista] Casali, Maria Rita; Grasselli, Luigi abstract For each integer g>1, a class$M_g$of “2-symmetric” crystallizations, depending on a 2(g+1)-tuple of positive integers satisfying simple conditions is introduced; the “2-symmetry” implies that the represented closed, orientable 3-manifolds are 2-fold covering spaces of$S^3$branched over a link. Since every closed, orientable 3-manifold M of Heegaard genus$g \le 2$admits a crystallization belonging to$M_g$, we obtain an easy proof og the fact that M is a 2-fold covering spaces of$S^3$branched over a link. Further, the class contains all Lins-Mandel crystallizations S(b,l,t,c), with l odd, which are thus proved to represent 2-fold branched coverings of$S^3$. 1990 - Una caratterizzazione dei corpi di manici 3-dimensionali [Articolo su rivista] Casali, Maria Rita abstract In this work we produce the following characterization of 3-dimensional "handlebodies": a 3-manifold is a handlebody if and only if its regular genus equals the genus of its boundary. 1990 - Wave moves on crystallizations [Articolo su rivista] Casali, Maria Rita; Grasselli, Luigi abstract In this paper, the relations between the notions of “wave move” (by Homma-Ochiai) and “frame” (by Tsukui) are investigated. A genus three frame of$S^3$is produced, giving a counterexample to a conjecture of Tsukui; on the contrary, the conjecture is proved to be true in genus two. 1989 - A catalogue of the genus two 3-manifolds [Articolo su rivista] Casali, Maria Rita abstract In this paper, the extension of the notion of “wave move” to crystallizations (by Casali-Grasselli) leads to a “reduced” catalogue of all genus$g\le 2$3-manifolds, depending on 6-tuples of positive integers. The appendix shows a partial output of the computer program which generates the catalogue and gives a presentation of the fundamental group of each element. 1989 - Representing branched coverings by edge coloured graphs [Articolo su rivista] Casali, Maria Rita; Grasselli, Luigi abstract Given a link L in$S^3$, we describe a standard method for constructing a class$\Gamma_{L,d}$of 4-coloured graphs representing all closed orientable 3-manifolds which are d-fold coverings of$S^3$branched over the link L. 1988 - Characterizing crystallizations among Lins-Mandel 4-coloured graphs [Articolo su rivista] Casali, Maria Rita; Grasselli, Luigi abstract In [Discr. Math. 57 (1985), 261-284], Lins and Mandel introduce a class of 3-manifolds represented by 4-coloured graphs S(b,l,t,c) depending on a 4-tuple (b,l,t,c) of positive integers; moreover, they prove that, if the following conditions hold$(b,c)=1$,$(l,t)=1$,$c=(-1)^t$if l odd, then S(b,l,t,c) is a crystallization of an (orientable) 3-manifold.In this paper we show that the above conditions are also necessary: hence, they characterize crystallizations among Lins-Mandel graphs. 1987 - Fundamental groups of branched covering spaces of S^3 [Articolo su rivista] Casali, Maria Rita abstract Given a knot K in$S^3$, it is known a standard method (by Casali and Grasselli) for constructing a 4-coloured graph representing the closed orientable 3-manifold$M=M(K,d,\omega)$which is the d-fold covering space of$S^3$branched over K and associated to the transitive d-representation$\omega$of the knot group. In this paper we obtain a presentation of the fundamental group of M, directly from the Wirtinger presentation of the knot group and from the transitive d-representation$\omega$. 1987 - LE 3-VARIETA' COME TRIPLI RIVESTIMENTI RAMIFICATI DI S^3 [Abstract in Atti di Convegno] Casali, Maria Rita abstract E' nota la possibilità di rappresentare le n-varietà PL attraverso particolari grafi colorati sugli spigoli mediante n+1 colori, detti cristallizzazioni (si veda [M.Pezzana, Sulla struttura topologica delle varietà compatte, Atti Sem. Mat. Fis. Univ. Modena 23 (1974), 269-277] o [M.Ferri - C.Gagliardi - L.Grasselli, A graph--theoretical representation of PL-manifolds. A survey on crystallizations, Aequationes Math. 31 (1986), 121-141]). In questo lavoro si presenta un approccio, basato esclusivamente sulla teoria delle cristallizzazioni, alla dimostrazione del seguente teorema (originariamente provato da Hilden e Montesinos): Ogni 3-varietà chiusa e orientabile M è triplo rivestimento semplice di$S^3\$ ramificato su un nodo K.