Compact 3-manifolds via 4-colored graphs

Cristofori, Paola; Mulazzani, Michele

doi:10.1007/s13398-015-0240-8

Cited by 11 publications

(20 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If Γ represents a closed n-manifold all dipoles of Γ are proper and Casali proved in [3] that dipole moves are sufficient to connect different 4-colored graphs representing the same closed 3manifold. This result is no longer true in the case with boundary, even in dimension three (see [15]).…”

Section: Dipole Movesmentioning

confidence: 97%

See 1 more Smart Citation

Compactn-Manifolds via (n+ 1)-Colored Graphs: a New Approach

Grasselli

Mulazzani

2020

Algebra Colloq.

Self Cite

View full text Add to dashboard Cite

We introduce a representation via (n + 1)-colored graphs of compact n-manifolds with (possibly empty) boundary, which appears to be very convenient for computer aided study and tabulation. Our construction is a generalization to arbitrary dimension of the one recently given by Cristofori and Mulazzani in dimension three, and it is dual to the one given by Pezzana in the seventies. In this context we establish some results concerning the topology of the represented manifolds: suspension, fundamental groups, connected sums and moves between graphs representing the same manifold. Classification results of compact orientable 4-manifolds representable by graphs up to six vertices are obtained, together with some properties of the G-degree of 5-colored graphs relating this approach to tensor models theory.

show abstract

Section: Dipole Movesmentioning

confidence: 97%

“…For ∆ = {i, j} we use the simplified notation g i,j instead of g {i,j} . 2 Such type of graphs were called contracted in[20] and in related subsequent papers, but in[15] the term contracted refers to a more general class of colored graphs.…”

mentioning

confidence: 99%

Compactn-Manifolds via (n+ 1)-Colored Graphs: a New Approach

Grasselli

Mulazzani

2020

Algebra Colloq.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Therefore, it seems to be fruitful in this framework to look for classifications results concerning all pseudomanifolds, or at least singular manifolds (subsection 6.1). The recently introduced representation theory for 3-manifolds with boundary (and their naturally associated singular manifolds) via regular 4-colored graphs (see [21]), if suitably extended to higher dimensions, might be a significant tool for this purpose.…”

Section: Conclusion and Research Trendsmentioning

confidence: 99%

Topology in colored tensor models via crystallization theory

Casali

Cristofori

Dartois

et al. 2018

Journal of Geometry and Physics

Self Cite

View full text Add to dashboard Cite

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.\ud 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.\ud 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.\ud 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

show abstract

“…The extension of the representation by 4-colored graphs to 3-manifolds with boundary has been performed in [8], where any 4-colored graph is associated to a compact 3-manifold with (possibly empty) boundary without spherical components, this correspondence being surjective on the whole class of such manifolds.…”

Section: Introductionmentioning

confidence: 99%

4-colored Graphs and Knot/Link Complements

et al. 2017

Self Cite

View full text Add to dashboard Cite

Abstract. A representation for compact 3-manifolds with non-empty non-spherical 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 classification 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/classification 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.2010 Mathematics Subject Classification: 57M25, 57N10, 57M15.

show abstract

Compact 3-manifolds via 4-colored graphs

Cited by 11 publications

References 25 publications

Compactn-Manifolds via (n+ 1)-Colored Graphs: a New Approach

Compactn-Manifolds via (n+ 1)-Colored Graphs: a New Approach

Topology in colored tensor models via crystallization theory

4-colored Graphs and Knot/Link Complements

Contact Info

Product

Resources

About