4-colored Graphs and Knot/Link Complements

Cristofori, Paola; Fominykh, Evgeny; Mulazzani, Michele; Tarkaev, V. V.

doi:10.1007/s00025-017-0686-4

Cited by 9 publications

(12 citation statements)

References 15 publications

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“…In [11], all prime orientable 3-manifolds with toric boundary representable by (bipartite) 4-colored graphs with order ≤ 12 have been classified.…”

Section: Manifolds Of Graph Complexity 14mentioning

confidence: 99%

“…The classification has been obtained starting from the catalogues of graphs described in [11] by using the programs 3-Manifold Recognizer [16] and SnapPy [12] and following the procedure described in the same paper. Theorem 6.…”

Section: Manifolds Of Graph Complexity 14mentioning

confidence: 99%

“…On the other hand, it follows from Lemma 1 that 3] that Γ represents the tetrahedral manifold otet12_00009, which is obtained by gluing together 12 regular ideal tetrahedra (see details in [11]). The conclusion c g (M G ) = 12n now follows from Theorem 3.…”

Section: Exact Values and Two-sided Bounds For The Graph Complexity O...mentioning

confidence: 99%

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Minimal 4-colored graphs representing an infinite family of hyperbolic 3-manifolds

Cristofori

Fominykh

Mulazzani

et al. 2017

RACSAM

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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.2010 Mathematics Subject Classification: 57N10, 57Q15, 57M15.

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“…In [11], all prime orientable 3-manifolds with toric boundary representable by (bipartite) 4-colored graphs with order ≤ 12 have been classified.…”

Section: Manifolds Of Graph Complexity 14mentioning

confidence: 99%

Section: Manifolds Of Graph Complexity 14mentioning

confidence: 99%

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Minimal 4-colored graphs representing an infinite family of hyperbolic 3-manifolds

Cristofori

Fominykh

Mulazzani

et al. 2017

RACSAM

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“…Therefore, as further development, we have restricted our attention to the study of orientable manifolds with (possibly disconnected) toric boundary. In a forthcoming paper ( [14]), we have completed the classification of the manifolds involved in Proposition 21: all of them turn out to be complement of knots or links in S 3 . In particular, the two elements of C (6) represent the complements of the Hopf link and the trivial knot, while the manifolds M Γ 1 , M Γ 2 , M Γ 3 of the above proposition are complements of the links L8n7, L6n1 and L8n8 respectively (notations are according to Thistlethwaite Link Table ).…”

Section: Complexitymentioning

confidence: 99%

Compact 3-manifolds via 4-colored graphs

Cristofori

Mulazzani

2015

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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 eighties 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).

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“…The following theorem extends to singular manifolds a well-known result -due to Pezzana ([17])founding the combinatorial representation theory for closed PL-manifolds of arbitrary dimension via colored graphs (the so called crystallization theory). See also [11] and [12] for the 3-dimensional case.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial properties of the G-degree

Casali

Grasselli

2018

Rev Mat Complut

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

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4-colored Graphs and Knot/Link Complements

Cited by 9 publications

References 15 publications

Minimal 4-colored graphs representing an infinite family of hyperbolic 3-manifolds

Minimal 4-colored graphs representing an infinite family of hyperbolic 3-manifolds

Compact 3-manifolds via 4-colored graphs

Combinatorial properties of the G-degree

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