
Paola CRISTOFORI
Professore Associato Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede exMatematica

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2024
 Geminduced trisections of compact PL 4manifolds
[Articolo su rivista]
Casali, M. R.; Cristofori, P.
abstract
The idea of studying trisections of closed smooth 4manifolds via (singular) triangulations, endowed with a suitable vertexlabelling by three colors, is due to Bell, Hass, Rubinstein and Tillmann, and has been applied by Spreer and Tillmann to standard simplyconnected 4manifolds, via the socalled simple crystallizations. In the present paper we propose generalizations of these ideas by taking into consideration a possible extension of trisections to compact PL 4manifolds with connected boundary, which is related to Birman's special Heegaard sewing, and by analyzing geminduced trisections, i.e. trisections that can be induced not only by simple crystallizations, but also by any 5colored graph encoding a PL 4manifold with empty or connected boundary. This last notion gives rise to that of Gtrisection genus, as an analogue, in this context, of the wellknown trisection genus. We give conditions on a 5colored graph ensuring one of its geminduced trisections  if any  to realize the Gtrisection genus, and prove how to determine it directly from the graph. As a consequence, we detect a class of closed simplyconnected 4manifolds, comprehending all standard ones, for which both Gtrisection genus and trisection genus coincide with the second Betti number and also with half the value of the graphdefined PL invariant regular genus. Moreover, the existence of geminduced trisections and an estimation of the Gtrisection genus via surgery description is obtained, for each compact PL 4manifold admitting a handle decomposition lacking in 3handles.
2023
 Classifying compact 4manifolds via generalized regular genus and Gdegree
[Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola
abstract
$(d+1)$colored graphs, i.e. edgecolored 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 Gdegree: the first one extending to higher dimension the classical notion of Heegaard genus for 3manifolds, 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 Gdegree 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 Gdegree at most 18; moreover, in case of empty or connected boundary, the classifications are extended to generalized regular genus two and to Gdegree 24.
2023
 Kirby diagrams and 5colored graphs representing compact 4manifolds
[Articolo su rivista]
Casali, M. R.; Cristofori, P.
abstract
It is wellknown that in dimension 4 any framed link (L, c) uniquely represents the PL 4manifold M^4 (L, c) obtained from D^4 by adding 2handles along (L, c). Moreover, if trivial dotted components are also allowed (i.e. in case of a Kirby diagram (L^(*), d)), the associated PL 4manifold M^4(L^(*), d) is obtained from D^4 by adding 1handles along the dotted components and 2handles 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 edgecolored graphs: in particular, we describe how to construct algorithmically a (regular) 5colored 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 (simplyconnected) PL 4manifold admitting handle decompositions without 3handles. Furthermore, upper bounds for both the invariants gemcomplexity and regular genus of M^4(L^(*), d) are obtained, in terms of the combinatorial properties of the Kirby diagram.
2023
 TOPOLOGY IN COLORED TENSOR MODELS
[Poster]
Casali, Maria Rita; Cristofori, Paola; Grasselli, Luigi
abstract
From a “geometric topology” point of view, the theory of manifold representation by means of edgecolored 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 dmanifold by means of a totally combinatorial tool.
Edgecolored 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 Gdegree 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 twodimensional matrix model setting.
Therefore, topological and geometrical properties of the represented PL manifolds, with respect to the Gdegree, have speciﬁc relevance in the tensor models framework, showing a direct fruitful interaction between tensor models and discrete geometry, via edgecolored graphs.
In colored tensor models, manifolds and pseudomanifolds are (almost) on the same footing, since they constitute the class of polyhedra represented by edgecolored Feynman graphs arising in this context; thus, a promising research trend is to look for classiﬁcation results concerning all pseudomanifolds represented by graphs of a given Gdegree. In dimension 4, the goal has already been achieved  via singular 4manifolds  for all compact PL 4manifolds with connected boundary up to Gdegree 24.
In the same dimension, the existence of colored graphs encoding different PL manifolds with the same underlying TOP manifold, suggests also to investigate the ability of tensor models to accurately reﬂect geometric degrees of freedom of Quantum Gravity.
2021
 Compact 4manifolds 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 1handles or in 1 and 3handles, thus facing also the problem, posed by Kirby, of the existence of {em special handlebody decompositions} for any simplyconnected closed PL $4$manifold. In particular, we detect a class of compact simplyconnected 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 simplyconnected PL $4$manifold with empty or connected boundary having colored triangulations that minimize the combinatorially defined PL invariants {em regular genus, gemcomplexity} or {em Gdegree} among all such manifolds with the same second Betti number.
2020
 Crystallizations of compact 4manifolds minimizing combinatorially defined PLinvariants
[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 semisimple and weak semisimple crystallizations, with a particular attention to their properties of minimizing combinatorially defined PLinvariants, such as the regular genus, the Gurau degree, the gemcomplexity and the (geminduced) 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 semisimple crystallizations.
2018
 Gdegree for singular manifolds
[Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola; Grasselli, Luigi
abstract
The Gdegree of colored graphs is a key concept in the approach to Quantum Gravity via tensor models.
The present paper studies the properties of the Gdegree for the large class of graphs representing singular manifolds (including closed PL manifolds).
In particular, the complete topological classification up to Gdegree 6 is obtained in dimension 3, where all 4colored graphs represent singular manifolds.
2018
 Minimal 4colored graphs representing an infinite family of hyperbolic 3manifolds
[Articolo su rivista]
Cristofori, Paola; Fominykh, Evgeny; Mulazzani, Michele; Tarkaev, Vladimir
abstract
The graph complexity of a compact 3manifold is defined as the minimum order among all 4colored 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 3manifolds 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 edgecolored 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 dmanifold by means of a totally combinatorial tool.
Edgecolored 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 Gdegree 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 twodimensional matrix model setting.
Therefore, topological and geometrical properties of the represented PL manifolds, with respect to the Gdegree, have specific relevance in the tensor models framework, show ing a direct fruitful interaction between tensor models and discrete geometry, via edgecolored graphs.
In colored tensor models, manifolds and pseudomanifolds are (almost) on the same footing, since they constitute the class of polyhedra represented by edgecolored Feynman graphs arising in this context; thus, a promising research trend is to look for classification results concerning all pseudomanifolds  or, at least, singular dmanifolds, if d ≥ 4  represented by graphs of a given Gdegree.
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 PLmanifolds representation by means of edgecolored graphs (crystallization theory). On the other hand, the core of the paper is to establish results about the topological and geometrical properties of the Guraudegree (or Gdegree) of the represented manifolds, in relation with the motivations coming from physics.
In fact, the Gdegree 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 twodimensional matrix model setting.
In particular, the Gdegree of PLmanifolds is proved to be finitetoone in any dimension, while in dimension 3 and 4 a series of classification theorems are obtained for PLmanifolds represented by graphs with a fixed Gdegree.
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
 4colored graphs and knot/link complements
[Articolo su rivista]
Cristofori, Paola; Fominykh, Evgeny; Mulazzani, Michele; Tarkaev, Vladimir
abstract
A representation for compact 3manifolds with nonempty nonspherical boundary via 4colored graphs (i.e., 4regular graphs endowed with a proper edgecoloration 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 3manifolds with toric boundary, which contains the important case of complements of knots and links in the 3sphere.
In this paper we obtain the complete catalogation/classication of these 3manifolds 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 socalled natural doubles which are: the complex double, the Schottky double and the orienting double. We prove that if X is a nonorientable Klein surface with
nonempty 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 4manifolds via crystallizations: results and open problems
[Capitolo/Saggio]
Casali, Maria Rita; Cristofori, Paola; Gagliardi, Carlo
abstract
Crystallization theory is a graphtheoretical representation method for compact PLmanifolds of arbitrary dimension, which makes use of a particular class of edgecoloured 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 graphdefined invariants for PLmanifolds.
The present survey paper focuses on the 4dimensional case, presenting uptodate results about the PL classication of closed 4manifolds, by means of two such PL invariants: regular genus and gemcomplexity.
Open problems are also presented, mainly concerning different classication of 4manifolds in TOP and DIFF=PL categories, and a possible approach to the 4dimensional Smooth Poincare Conjecture.
2016
 Compact 3manifolds via 4colored graphs
[Articolo su rivista]
Cristofori, Paola; Mulazzani, Michele
abstract
We introduce a representation of compact 3manifolds without spherical boundary components via (regular) 4colored 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 3manifolds, 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 3manifolds representable by graphs with few vertices (≤6 in the nonorientable case and ≤8 in the orientable one).
2016
 PL 4manifolds admitting simple crystallizations: framed links and regular genus
[Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola; Gagliardi, Carlo
abstract
Simple crystallizations are edgecolored graphs representing piecewise linear (PL) 4manifolds with the property that the 1skeleton of the associated triangulation equals the 1skeleton of a 4simplex. In this paper, we prove that any (simplyconnected) PL 4manifold 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 4manifold by k(M)=3β_2(M), where k(M) is the gemcomplexity of M (i.e. the nonnegative number p−1, 2p being the minimum order of a crystallization of M), implies that both PL invariants gemcomplexity and regular genus turn out to be additive within the class of all PL 4manifolds admitting simple crystallizations (in particular, within the class of all “standard” simplyconnected PL 4manifolds).
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 3manifold can be estimated by means of the combinatorial notion of GMcomplexity. In this paper, we prove that the GMcomplexity 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 3manifolds turn out to exist, where complexity and GMcomplexity 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 3manifold M and k(M)+1 is half the minimum order of a crystallization of M
2015
 Cataloguing PL 4manifolds by gemcomplexity
[Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola
abstract
We describe an algorithm to subdivide automatically a given set of PL nmanifolds (via coloured triangulations or, equivalently, via crystallizations) into classes whose elements are PLhomeomorphic. The algorithm, implemented in the case n = 4, succeeds to solve completely the PLhomeomorphism problem among the catalogue of all closed connected PL 4manifolds up to gemcomplexity 8 (i.e.,
which admit a coloured triangulation with at most 18 4simplices).
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 nonexistence of exotic PL 4manifolds up to gemcomplexity 8 is proved. Further applications of the tool are described, related to possible PLrecognition of different triangulations of the K3surface.
2013
 A code for disconnected edgecolored graphs
[Articolo su rivista]
Cristofori, Paola
abstract
We extend the definition of code of an edgecolored graph, given in [S. Lins, Gems, computers and attractors for3manifolds, Knots and Everything, 5, World Scientific,1995] and [M.R. Casali  C. Gagliardi, A code for mbipartite edgecoloured graphs, Rend. Ist. Mat. Univ. Trieste, 32, suppl. 1, (2001), 5576], to the disconnected case and prove that our code keeps the same property of detecting colorisomorphic graphs.
2013
 Coloured graphs representing PL 4manifolds
[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 edgecoloured graphs. The combinatorial nature of this representation allows to elaborate and implement algorithmic procedures for the generation and analysis of catalogues of closed PL nmanifolds. 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 GemMatveev complexity (GMcomplexity) has been introduced within crystallization theory, as a combinatorial method to estimate Matveev's complexity of closed 3manifolds; it yielded upper bounds for interesting classes of such manifolds. In this paper, we extend the definition to the case of nonempty boundary and prove that for each compact irreducible and boundaryirreducible 3manifold it coincides with the modified Heegaard complexity introduced by Cattabriga, Mulazzani and Vesnin. Moreover, via GMcomplexity, we obtain an estimation of Matveev's complexity for all Seifert 3manifolds with base D2 and two exceptional fibers and, therefore, for all torus knot complements.
2013
 Gammaclass_4dim: A program to subdivide a set of rigid crystallizations of closed 4manifolds into equivalence classes, whose elements represent PLhomeomorphic manifolds.
[Software]
Casali, Maria Rita; Cristofori, Paola
abstract
Gammaclass_4dim is a program yielding, from any given list X of crystallizations of 4dimensional PLmanifolds, the automatic partition of the elements of X into equivalence classes, such that each class contains only crystallizations representing the same PLmanifold. Moreover, the program attempts the identification of the represented 4manifolds by means of comparison of the representatives of each class with known catalogues of crystallizations and/or by means of splitting into connected sums.
Gammaclass_4dim is based on the existence of elementary combinatorial moves available for crystallizations of PLmanifolds of any dimension (i.e. the wellknown "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), 10011035].
The program has already been tested for known catalogues of crystallizations of 4manifolds, by making use of a fixed admissible sequence of the above moves; further applications are in progress.
2013
 Generation of Catalogues of PL nmanifolds: 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 piecewiselinear nmanifolds, i.e.
triangulable spaces which appear locally as the ndimensional Euclidean space. This paper reports on the computational aspects of an algorithm for generating triangulations of PL 3 and 4manifolds represented by edgecoloured 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 multicore 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 3manifolds 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 3manifolds via crystallization theory, yielding the notion of GemMatveev complexity; the other one for compact orientable 3manifolds via generalized Heegaard diagrams, yielding the notion of modified Heegaard complexity. In this paper we extend to the nonorientable case the definition of modified Heegaard complexity and prove that for closed 3manifolds GemMatveev 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 threedimensional 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 3component links.
2012
 Review about "A note on Gornik's perturbation of KhovanovRozansky 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 KhovanovRozansky 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 BarNatan'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 3braids" 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 3manifolds
[Articolo su rivista]
Bandieri, Paola; Casali, Maria Rita; Cristofori, Paola; Grasselli, Luigi; M., Mulazzani
abstract
The present paper is a survey of uptodate results in 3dimensional crystallization theory, in particular along the following directions: generation and analysis of catalogues of PLmanifolds for increasing values of the vertex number of the representing graphs; definition and/or computation of invariants for PLmanifolds, directly from the representing graphs.In particular, with regard to PLmanifold 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 3varietà: 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 GemMatveev complessità per 3varietà chiuse.
2010
 A census of genustwo 3manifolds up to 42 coloured tetrahedra
[Articolo su rivista]
Bandieri, Paola; Cristofori, Paola; Gagliardi, Carlo
abstract
We improve and extend to the nonorientable case a recent result of Karábaš, Maličký and Nedela concerning the classification of all orientable prime 3manifolds of Heegaard genus two, triangulated with at most 42 coloured tetrahedra.
2010
 Review about "A remark on Khovanov homology and twofold 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 Gequivariant 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 60th 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
 Nonorientable 3manifolds admitting colored triangulations with at most 30 tetrahedra.
[Articolo su rivista]
Bandieri, Paola; Cristofori, Paola; Gagliardi, Carlo
abstract
We present the census of all nonorientable, closed, connected 3manifolds 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 nonbipartite crystallizations up to 30 vertices.
2009
 Review about "A computation in KhovanovRozansky homology" by D. Krasner
[Recensione in Rivista]
Cristofori, Paola
abstract
2009
 Review about "A slice genus lower bound from sl(n) KhovanovRozansky 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 "Openclosed 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 quasialternating links" by A. Champanerkar, I. Kofman
[Recensione in Rivista]
Cristofori, Paola
abstract
2008
 A catalogue of orientable 3manifolds triangulated by 30 coloured tetrahedra
[Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola
abstract
The present paper follows the computational approach to 3manifold classification via edgecoloured graphs, already performed in [1] (with respect to orientable 3manifolds up to 28 coloured tetrahedra), in [2] (with respect to nonorientable3manifolds up to 26 coloured tetrahedra), in [3] and [4] (with respect to genus two 3manifolds up to 34 coloured tetrahedra): in fact, by automatic generation and analysis of suitable edgecoloured graphs, called crystallizations, we obtain a catalogue of all orientable 3manifolds 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 oneto one correspondence with the homeomorphism classes of the represented manifolds.
2008
 CRYSTALLIZATION CATALOGUES AND ARCHIVES OF CLOSED 3MANIFOLDS WITH LOW GEMCOMPLEXITY
[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 edgecoloured graphs representing all orientable and/or non orientable 3manifolds 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 nonorientable 3manifolds admitting decompositions into 26 coloured tetrahedra, Acta Appl. Math. 54 (1999), 7597]) 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 clusterless 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
 Gammaclass: A program to subdivide a set of rigid crystallizations of closed 3manifolds into equivalence classes, whose elements represent homeomorphic manifolds
[Software]
Casali, Maria Rita; Cristofori, Paola
abstract
Gammaclass is a program which implements the algorithm described in in [Casali M.R., Cristofori P., A catalogue of orientable 3manifolds triangulated by 30 coloured tetrahedra, Journal of Knot Theory and its Ramifications 17 (2008), no.5, 579599], 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 Gammaclass 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 multivariable 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 edgecoloured graphs representing PL ndimensional 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 nmanifolds, which has been performed by means of several functions, collected in a unified program, called DUKE III. DUKE III allows automatic manipulation of edgecoloured graphs representing PL nmanifolds (code computation, checking possible isomorphism between edgecoloured graphs, construction of boundary graph, checking bipartition, connectedness, rigidity and planarity conditions, combinatorial moves, invariants computation...). Furthermore, DUKE III allows automatic recognition of orientable 3manifolds triangulated by at most 30 coloured tetrahedra and of nonorientable 3manifolds triangulated by at most 26 coloured tetrahedra (by making use of existing electronic archives of all rigid bipartite crystallizations up to 30 vertices and nonbipartite 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_3link homology" by Mackaay, Marco and Vaz, Pedro
[Recensione in Rivista]
Cristofori, Paola
abstract
2007
 Stronglycyclic branched coverings of knots via (g,1)decompositions
[Articolo su rivista]
Cristofori, Paola; M., Mulazzani; A., Vesnin
abstract
Stronglycyclic 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 gwords 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 GMcomplexity introduced in [Casali, M.R., Topol. Its Appl., 144: 201209, 2004], the present paper performs a graphtheoretical approach to the computation of (Matveev's) complexity for closed orientable 3manifolds. In particular, the existing crystallization catalogue C28 available in [Lins, S., Knots and Everything 5, World Scientific, Singapore, 1995] is used to obtain upper bounds for the complexity of closed orientable 3manifolds 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 KaniaBartoszynska ideal" by Gilmer, Patrick M.
[Recensione in Rivista]
Cristofori, Paola
abstract
2006
 c_GM: A program to compute GMcomplexity of edgecoloured graphs representing closed 3manifolds
[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 nonorientable 3manifolds via crystallization theory, Topology and its Applications 144 (13) (2004), 201209], to estimate Matveev's complexity of a 3manifold starting from the code of an associated edgecoloured graph (GMcomplexity computation). This program has already allowed to compute GMcomplexity of all nonorientable 3manifolds represented by edgecoloured graphs up to 26 vertices (catalogue ~C26) and of all orientable 3manifolds represented by edgecoloured 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 nonorientable 3manifolds via crystallization theory, Topology and its Applications 144 (13) (2004), 201209] and [M.R. Casali  P.Cristofori, Computing Matveev's complexity via crystallization theory: the orientable case, Acta Applicandae Mathematicae 92 (2006), 113123]. The program computes the GMcomplexity both of a single edgecoloured graph and of a list of edgecoloured graphs. It also computes the minimal GMcomplexity 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 edgecoloured 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 2variable 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 2strand 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 edgecoloured 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 3manifolds
[Articolo su rivista]
Cristofori, Paola
abstract
The Heegaard genus and the regular genus are two invariants for 3manifolds, which, as it is already known, coincide for orientable3manifolds with boundary. It is also known that the regular genus of a nonorientable closed 3manifold is simply twice its Heegaard genus.In this paper we prove that the same relations hold in the general case of compact 3manifolds.
1995
 Genere di Heegaard e genere regolare per 3varieta` 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 3varieta` 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 3varieta`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 3varieta` chiuse. La dimostrazione della loro coincidenza per il caso con bordo, nell'ipotesi di orientabilita`, utilizza risultati noti sugli insiemi universali di ramificazione per 3varieta`orientabili e tecniche combinatorie.
1995
 Heegaard and regular genus of 3manifolds 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 3manifold with boundary coincide.
1995
 Moves on coloured spines
[Articolo su rivista]
Bandieri, Paola; Cristofori, Paola
abstract
We define a set of combinatorial moves on 3coloured graphs representing spines of 3manifolds and study their effects on the crystallizations corresponding to the 3coloured 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 9vertex simplicial triangulation found by Banchoff and Kuhnel [The math. Intelligencer 53 (1983), 1122]; the second one is the contracted triangulation with eight 4simplexes, built by the third author [Aequationes Math. 37 (1989), 130140].