Nuova ricerca

Paola CRISTOFORI

Professore Associato
Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede ex-Matematica

Home | Curriculum(pdf) | Didattica |


Pubblicazioni

2024 - Gem-induced trisections of compact PL 4-manifolds [Articolo su rivista]
Casali, M. R.; Cristofori, P.
abstract

The idea of studying trisections of closed smooth 4-manifolds via (singular) triangulations, endowed with a suitable vertex-labelling by three colors, is due to Bell, Hass, Rubinstein and Tillmann, and has been applied by Spreer and Tillmann to standard simply-connected 4-manifolds, via the so-called simple crystallizations. In the present paper we propose generalizations of these ideas by taking into consideration a possible extension of trisections to compact PL 4-manifolds with connected boundary, which is related to Birman's special Heegaard sewing, and by analyzing gem-induced trisections, i.e. trisections that can be induced not only by simple crystallizations, but also by any 5-colored graph encoding a PL 4-manifold with empty or connected boundary. This last notion gives rise to that of G-trisection genus, as an analogue, in this context, of the well-known trisection genus. We give conditions on a 5-colored graph ensuring one of its gem-induced trisections - if any - to realize the G-trisection genus, and prove how to determine it directly from the graph. As a consequence, we detect a class of closed simply-connected 4-manifolds, comprehending all standard ones, for which both G-trisection genus and trisection genus coincide with the second Betti number and also with half the value of the graph-defined PL invariant regular genus. Moreover, the existence of gem-induced trisections and an estimation of the G-trisection genus via surgery description is obtained, for each compact PL 4-manifold admitting a handle decomposition lacking in 3-handles.

2023 - 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.

2023 - Kirby diagrams and 5-colored graphs representing compact 4-manifolds [Articolo su rivista]
Casali, M. R.; Cristofori, P.
abstract

It is well-known that in dimension 4 any framed link (L, c) uniquely represents the PL 4-manifold M^4 (L, c) obtained from D^4 by adding 2-handles along (L, c). Moreover, if trivial dotted components are also allowed (i.e. in case of a Kirby diagram (L^(*), d)), the associated PL 4-manifold M^4(L^(*), d) is obtained from D^4 by adding 1-handles along the dotted components and 2-handles along the framed components. In this paper we study the relationships between framed links and/or Kirby diagrams and the representation theory of compact PL manifolds by edge-colored graphs: in particular, we describe how to construct algorithmically a (regular) 5-colored graph representing M^4(L^(*), d), directly "drawn over" a planar diagram of (L^(*), d), or equivalently how to algorithmically obtain a triangulation of M^4(L^(*),d). As a consequence, the procedure yields triangulations for any closed (simply-connected) PL 4-manifold admitting handle decompositions without 3-handles. Furthermore, upper bounds for both the invariants gem-complexity and regular genus of M^4(L^(*), d) are obtained, in terms of the combinatorial properties of the Kirby diagram.

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 - 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.

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 - Minimal 4-colored graphs representing an infinite family of hyperbolic 3-manifolds [Articolo su rivista]
Cristofori, Paola; Fominykh, Evgeny; Mulazzani, Michele; Tarkaev, Vladimir
abstract

The graph complexity of a compact 3-manifold is defined as the minimum order among all 4-colored graphs representing it. Exact calculations of graph complexity have been already performed, through tabulations, for closed orientable manifolds (up to graph complexity 32) and for compact orientable 3-manifolds with toric boundary (up to graph complexity 12) and for infinite families of lens spaces. In this paper we extend to graph complexity 14 the computations for orientable manifolds with toric boundary and we give two sided bounds for the graph complexity of tetrahedral manifolds. As a consequence, we compute the exact value of this invariant for an infinite family of such 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 - 4-colored graphs and knot/link complements [Articolo su rivista]
Cristofori, Paola; Fominykh, Evgeny; Mulazzani, Michele; Tarkaev, Vladimir
abstract

A representation for compact 3-manifolds with non-empty nonspherical boundary via 4-colored graphs (i.e., 4-regular graphs endowed with a proper edge-coloration with four colors) has been recently introduced by two of the authors, and an initial classication of such manifolds has been obtained up to 8 vertices of the representing graphs. Computer experiments show that the number of graphs/manifolds grows very quickly as the number of vertices increases. As a consequence, we have focused on the case of orientable 3-manifolds with toric boundary, which contains the important case of complements of knots and links in the 3-sphere. In this paper we obtain the complete catalogation/classication of these 3-manifolds up to 12 vertices of the associated graphs, showing the diagrams of the involved knots and links. For the particular case of complements of knots, the research has been extended up to 16 vertices.

2017 - The double of the doubles of Klein surfaces [Articolo su rivista]
Costa, Antonio F.; Cristofori, Paola; Porto, Ana M.
abstract

A Klein surface is a surface with a dianalytic structure. A double of a Klein surface X is a Klein surface Y such that there is a degree two morphism (of Klein surfaces) Y -> X. There are many doubles of a given Klein surface and among them the so-called natural doubles which are: the complex double, the Schottky double and the orienting double. We prove that if X is a non-orientable Klein surface with non-empty boundary, the three natural doubles, although distinct Klein surfaces, share a common double: "the double of doubles" denoted by DX. We describe how to use the double of doubles in the study of both moduli spaces and automorphisms of Klein surfaces. Furthermore, we show that the morphism from DX to X is not given by the action of an isometry group on classical surfaces.

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 - Compact 3-manifolds via 4-colored graphs [Articolo su rivista]
Cristofori, Paola; Mulazzani, Michele
abstract

We introduce a representation of compact 3-manifolds without spherical boundary components via (regular) 4-colored graphs, which turns out to be very convenient for computer aided study and tabulation. Our construction is a direct generalization of the one given in the 1980s by S. Lins for closed 3-manifolds, which is in turn dual to the earlier construction introduced by Pezzana’s school in Modena. In this context we establish some results concerning fundamental groups, connected sums, moves between graphs representing the same manifold, Heegaard genus and complexity, as well as an enumeration and classification of compact 3-manifolds representable by graphs with few vertices (≤6 in the non-orientable case and ≤8 in the orientable one).

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.

2013 - A code for disconnected edge-colored graphs [Articolo su rivista]
Cristofori, Paola
abstract

We extend the definition of code of an edge-colored graph, given in [S. Lins, Gems, computers and attractors for3-manifolds, Knots and Everything, 5, World Scientific,1995] and [M.R. Casali - C. Gagliardi, A code for m-bipartite edge-coloured graphs, Rend. Ist. Mat. Univ. Trieste, 32, suppl. 1, (2001), 55-76], to the disconnected case and prove that our code keeps the same property of detecting color-isomorphic graphs.

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 - Generation of Catalogues of PL n-manifolds: Computational Aspects on HPC Systems [Articolo su rivista]
Alessandro, Marani; Marzia, Rivi; Cristofori, Paola
abstract

Within mathematical research, Geometric Topology deals with the study of piecewise-linear n-manifolds, i.e. triangulable spaces which appear locally as the n-dimensional Euclidean space. This paper reports on the computational aspects of an algorithm for generating triangulations of PL 3- and 4-manifolds represented by edge-coloured graphs. As the number of graph vertices is increased the algorithm becomes computationally expensive very quickly, making it a natural candidate for the usage of HPC resources. We present an optimized, parallel version of the algorithm that is suitable for deployment of multi-core systems. Scalability results are discussed on two different platforms, namely an IBM iDataPlex Linux cluster and the IBM supercomputer BlueGene/Q.

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.

2012 - Cyclic generalizations of two hyperbolic icosahedral manifolds [Articolo su rivista]
Cristofori, Paola; T., Kozlovskaya; A., Vesnin
abstract

We study two families of closed orientable three-dimensional manifolds, which are defined as cyclic generalizations of two hyperbolic icosahedral manifolds, which were described first by Richardson and Rubinstein and then by Everitt. Results about covering properties, fundamental groups and hyperbolic volumes are proved for the manifolds belonging to these families. In particular, we show that they are cyclic coverings of the lens space L(3,1) branched over some 2- or 3-component links.

2012 - Review about "A note on Gornik's perturbation of Khovanov-Rozansky homology" by A. Lobb [Recensione in Rivista]
Cristofori, Paola
abstract

In the paper under review, the author starts from a spectral sequence defined by B. Gornik for Khovanov-Rozansky homology. The graded complex vector space H~i,jn(K) associated to Gornik's spectral sequence is supported in homological degree zero. The author shows that the quantum degrees of the nonzero H~0,jn(K) are determined only by an even integer sn(K). As a consequence sn(K) provides a lower bound for the smooth slice genus of K.

2012 - Review about "Surfaces with pulleys and Khovanov homology" by Audoux B. [Recensione in Rivista]
Cristofori, Paola
abstract

In the paper under review the author defines a refinement of Bar-Natan's construction of Khovanov homology, based on surfaces with pulleys and inducing known and new invariants for links in orientable surfaces.

2012 - Review about "The Khovanov width of twisted links and closed 3-braids" by Lowrance A. [Recensione in Rivista]
Cristofori, Paola
abstract

The paper under review deals with the problem of determining the support of the Khovanov homology of a link.

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 - Review about "Topological quantum information, Khovanov homology and the Jones polynomial" by Kauffman L.H. [Recensione in Rivista]
Cristofori, Paola
abstract

In the paper under review the author gives a quantum statistical interpretation for the bracket and the Jones polynomial of a link.

2011 - Stime della complessità di Matveev di una 3-varietà: diagrammi di Heegaard generalizzati e grafi colorati. [Abstract in Atti di Convegno]
Cristofori, Paola
abstract

Nella comunicazione si presenta l'estensione al caso non orientabile della definizione di complessità di Heegaard modificata e si dimostra la sua coincidenza con la Gem-Matveev complessità per 3-varietà chiuse.

2010 - A census of genus-two 3-manifolds up to 42 coloured tetrahedra [Articolo su rivista]
Bandieri, Paola; Cristofori, Paola; Gagliardi, Carlo
abstract

We improve and extend to the non-orientable case a recent result of Karábaš, Maličký and Nedela concerning the classification of all orientable prime 3-manifolds of Heegaard genus two, triangulated with at most 42 coloured tetrahedra.

2010 - Review about "A remark on Khovanov homology and two-fold branched covers" by L. Watson [Recensione in Rivista]
Cristofori, Paola
abstract

2010 - Review about "An oriented model for Khovanov homology" by C. Blanchet [Recensione in Rivista]
Cristofori, Paola
abstract

2010 - Review about "Equivalent Khovanov homology associated with symmetric links" by Chbili N. [Recensione in Rivista]
Cristofori, Paola
abstract

In the paper under review, the author defines, for each finite cyclic group G of odd order p, a G-equivariant Khovanov homology with coefficients in the field F2.

2010 - Review about "Signature, nullity and determinant of checkerboard colorable virtual links" by Im Y.H. - Lee K. - Lee S. [Recensione in Rivista]
Cristofori, Paola
abstract

The paper under review presents a generalization to checkerboard colorable virtual links of the definition of (modified) Goeritz matrix for a classical link in S3.

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)

2009 - Non-orientable 3-manifolds admitting colored triangulations with at most 30 tetrahedra. [Articolo su rivista]
Bandieri, Paola; Cristofori, Paola; Gagliardi, Carlo
abstract

We present the census of all non-orientable, closed, connected 3-manifolds admitting a rigid crystallization with at most 30 vertices. In order to obtain the above result, we generate, manipulate and compare, by suitable computer procedures, all rigid non-bipartite crystallizations up to 30 vertices.

2009 - Review about "A computation in Khovanov-Rozansky homology" by D. Krasner [Recensione in Rivista]
Cristofori, Paola
abstract

2009 - Review about "A slice genus lower bound from sl(n) Khovanov-Rozansky homology" by A. Lobb [Recensione in Rivista]
Cristofori, Paola
abstract

2009 - Review about "Fibred multilinks and singularities f\overline g" by A. Pichon and J. Seade [Recensione in Rivista]
Cristofori, Paola
abstract

2009 - Review about "Khovanov homology for signed divides" by O. Couture [Recensione in Rivista]
Cristofori, Paola
abstract

2009 - Review about "Notes on link homology" by M. Asaeda, M. Khovanov [Recensione in Rivista]
Cristofori, Paola
abstract

2009 - Review about "Open-closed TQFTS extend Khovanov homology from links to tangles" by A. Lauda, H. Pfeiffer [Recensione in Rivista]
Cristofori, Paola
abstract

2009 - Review about "Spanning trees and Khovanov homology" by A. Champanerkar and I. Kofman [Recensione in Rivista]
Cristofori, Paola
abstract

2009 - Review about "The Miyazawa polynomial of periodic virtual links" by J. Kim, S.Y. Lee, M. Seo [Recensione in Rivista]
Cristofori, Paola
abstract

2009 - Review about "Twisting quasi-alternating links" by A. Champanerkar, I. Kofman [Recensione in Rivista]
Cristofori, Paola
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 - Review about "On a background of the existence of multi-variable link invariants" by Nagasato, Fumikazu and Hamai, Kanau [Recensione in Rivista]
Cristofori, Paola
abstract

2008 - Review about "The quantum sl(3) invariants of cubic bipartite planar graphs" by Kim, Dongseok and Lee, Jaeun [Recensione in Rivista]
Cristofori, Paola
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 - Review about "Concordance crosscap number of a knot" by Zhang, Gengyu [Recensione in Rivista]
Cristofori, Paola
abstract

2007 - Review about "Meromorphic functions, bifurcation sets and fibred links" by Bodin, Arnaud and Pichon, Anne [Recensione in Rivista]
Cristofori, Paola
abstract

2007 - Review about "The universal sl_3-link homology" by Mackaay, Marco and Vaz, Pedro [Recensione in Rivista]
Cristofori, Paola
abstract

2007 - Strongly-cyclic branched coverings of knots via (g,1)-decompositions [Articolo su rivista]
Cristofori, Paola; M., Mulazzani; A., Vesnin
abstract

Strongly-cyclic branched coverings of knots are studied by using their (g,1)-decompositions. Necessary and sufficient conditions for the existence and uniqueness of such coverings are obtained.It is also shown that their fundamental groups admit geometric g-words cyclic presentations.

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.

2006 - Review about "A toy theory of Vassiliev invariants" by Duzhin, S. and Mostovoy, J. [Recensione in Rivista]
Cristofori, Paola
abstract

2006 - Review about "Magnetic graphs and an invariant for virtual links" by Miyazawa, Yasuyuki [Recensione in Rivista]
Cristofori, Paola
abstract

2006 - Review about "On the Frohman Kania-Bartoszynska ideal" by Gilmer, Patrick M. [Recensione in Rivista]
Cristofori, Paola
abstract

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.

2005 - Review about "A 2-variable polynomial invariant for a virtual link derived from magnetic graphs" by Kamada, Naoko and Miyazawa, Yasuyuki [Recensione in Rivista]
Cristofori, Paola
abstract

2005 - Review about "A family of knots yielding graph manifolds by Dehn surgery" by Yamada, Yuichi [Recensione in Rivista]
Cristofori, Paola
abstract

2005 - Review about "An endomorphism of the Khovanov invariant" by Lee, Eun Soo [Recensione in Rivista]
Cristofori, Paola
abstract

2005 - Review about "Constructing algebraic links for low edge numbers" by McCabe, Cynthia L. [Recensione in Rivista]
Cristofori, Paola
abstract

2005 - Review about "Virtual knots undetected by 1- and 2-strand bracket polynomials" by Dye, H. A. [Recensione in Rivista]
Cristofori, Paola
abstract

2004 - On the genus of S^m x S^n [Articolo su rivista]
Cristofori, Paola
abstract

By using a recursive algorithm, we construct edge-coloured graphs representing products of spheres and consequently we give upper bounds for the regular genus of S^m x S^n, for each m,n > 0.

2003 - Generalized regular genus for manifolds with boundary [Articolo su rivista]
Cristofori, Paola
abstract

We introduce a generalization of the regular genus, a combinatorial invariant of PL manifolds, which is proved to be strictly related, in dimension three, to the generalized Heegaard splittings defined by Montesinos.

2003 - Una generalizzazione delle varieta` di Dunwoody [Abstract in Atti di Convegno]
Cristofori, Paola
abstract

La comunicazione presenta una generalizzazione delle varieta`di Dunwoody attraverso la definizione di diagrammi di Heegaard di genere 2n, a simmetria ciclica di ordine n, che dipendono da un grafo a quattro vertici immerso in R^2-{(0,0)} e da cinque parametri interi soddisfacenti particolari condizioni.Si dimostra che le varieta`cosi`rappresentate sono rivestimenti fortemente ciclici di varieta`di genere due ramificati su (2,1)-nodi. Si da`una presentazione ciclica del gruppo fondamentale di tali varieta`indotta dal relativo diagramma di Heegaard.

1998 - Heegaard and regular genus agree for compact 3-manifolds [Articolo su rivista]
Cristofori, Paola
abstract

The Heegaard genus and the regular genus are two invariants for 3-manifolds, which, as it is already known, coincide for orientable3-manifolds with boundary. It is also known that the regular genus of a non-orientable closed 3-manifold is simply twice its Heegaard genus.In this paper we prove that the same relations hold in the general case of compact 3-manifolds.

1995 - Genere di Heegaard e genere regolare per 3-varieta` orientabili con bordo [Abstract in Atti di Convegno]
Cristofori, Paola
abstract

Il risultato oggetto della comunicazione e`la dimostrazione della coincidenza di due invarianti per 3-varieta` orientabili con bordo: il genere di Heegaard ed il genere regolare.Il primo e`un'estensione al caso con bordo del classico concetto di genere di Heegaard di una 3-varieta`chiusa. Il secondo e`un invariante PL che si configura come una generalizzazione a dimensione qualsiasi del concetto di genere di una superficie.E`gia`noto che i due invarianti coincidono per le 3-varieta` chiuse. La dimostrazione della loro coincidenza per il caso con bordo, nell'ipotesi di orientabilita`, utilizza risultati noti sugli insiemi universali di ramificazione per 3-varieta`orientabili e tecniche combinatorie.

1995 - Heegaard and regular genus of 3-manifolds with boundary [Articolo su rivista]
Cristofori, Paola; Gagliardi, Carlo; Grasselli, Luigi
abstract

By means of branched coverings techniques, we prove that the Heegaard genus and the regular genus of an orientable 3-manifold with boundary coincide.

1995 - Moves on coloured spines [Articolo su rivista]
Bandieri, Paola; Cristofori, Paola
abstract

We define a set of combinatorial moves on 3-coloured graphs representing spines of 3-manifolds and study their effects on the crystallizations corresponding to the 3-coloured graphs through the bijoin construction.

1993 - Linking two minimal triangulations of CP2 [Articolo su rivista]
R., Chiavacci; Cristofori, Paola; Gagliardi, Carlo
abstract

We present an explicit algorithm for linking two "minimal" triangulations of the complex projective plane. The first one is the 9-vertex simplicial triangulation found by Banchoff and Kuhnel [The math. Intelligencer 5-3 (1983), 11-22]; the second one is the contracted triangulation with eight 4-simplexes, built by the third author [Aequationes Math. 37 (1989), 130-140].