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Minimum ideal triangulations of hyperbolic 3-manifolds

Colin Adams, William Sherman
1991 Discrete & Computational Geometry  
Let a(n) be the minimum number of ideal hyperbolic tetrahedra necessary to construct a finite volume n-cusped hyperbolic 3-manifold, orientable or not.  ...  Let ~ror(n) be the corresponding number when we restrict ourselves to orientable manifolds. The correct values of a(n) and aor(n ) and the corresponding manifolds are given for n = 1, 2, 3, 4, and 5.  ...  We then collapse D down to a single edge as we shrink f to a point which becomes a new Minimum Ideal Triangulations of Hyperbolic 3-Manifolds 139 ideal vertex. In the process, we identify e to e'.  ... 
doi:10.1007/bf02574680 fatcat:rqe4jcpis5cpvfeq6bedmkmfcy

Poor ideal three-edge triangulations are minimal [article]

Evgeny Fominykh, Ekaterina Shumakova
2021 arXiv   pre-print
It is known that an ideal triangulation of a compact 3-manifold with nonempty boundary is minimal if and only if it contains the minimum number of edges among all ideal triangulations of the manifold.  ...  We exploit this property to construct minimal ideal triangulations for an infinite family of hyperbolic 3-manifolds with totally geodesic boundary.  ...  An ideal triangulation of a compact 3-manifold with nonempty boundary is minimal if and only if it contains the minimum number of edges among all ideal triangulations of the manifold.  ... 
arXiv:2105.05110v1 fatcat:hi3odnfhivf5llzvqcjekr4gli

Convex hulls and isometries of cusped hyperbolic 3-manifolds

Jeffrey R. Weeks
1993 Topology and its Applications  
An algorithm for computing canonical triangulations of cusped hyperbolic 3-manifolds provides an efficient way to determine whether two such manifolds are isometric.  ...  ., Convex hulls and isometries of cusped hyperbolic 3-manifolds, Topology and its Applications 52 (1993) 127-149.  ...  The main result of this paper is an algorithm which accepts as input an arbitrary ideal triangulation of a cusped hyperbolic 3-manifold, and produces as output the canonical triangulation of that manifold  ... 
doi:10.1016/0166-8641(93)90032-9 fatcat:kgx7gsjkhfga7g4z7cjjrfo52a

Complexity of 3-manifolds [article]

Bruno Martelli
2005 arXiv   pre-print
We give a summary of known results on Matveev's complexity of compact 3-manifolds. The only relevant new result is the classification of all closed orientable irreducible 3-manifolds of complexity 10.  ...  First, note that by the naturality property of the complexity c(M) is the minimum number of tetrahedra in an (ideal) triangulation. If M is closed, take a minimal triangulation T and straighten it.  ...  Among them, we find the Heegaard genus, the minimum number of tetrahedra in a triangulation, and Gromov's norm (which equals the volume when M is hyperbolic).  ... 
arXiv:math/0405250v2 fatcat:wq2ihz4lnrf6vkykwuz6whdxgq

The cusped hyperbolic census is complete [article]

Benjamin A. Burton
2014 arXiv   pre-print
For the first time, we prove here that the census meets its aim: we rigorously certify that every ideal 3-manifold triangulation with <= 8 tetrahedra is either (i) homeomorphic to one of the census manifolds  ...  From its creation in 1989 through subsequent extensions, the widely-used "SnapPea census" now aims to represent all cusped finite-volume hyperbolic 3-manifolds that can be obtained from <= 8 ideal tetrahedra  ...  As a final note: In the late 1980s, Adams and Sherman studied the minimum number of ideal tetrahedra required to build a k-cusped hyperbolic 3-manifold [1] .  ... 
arXiv:1405.2695v1 fatcat:swvvzg7odffzvjizwikhhbeycm

Algorithmic construction and recognition of hyperbolic 3-manifolds, links, and graphs [article]

Carlo Petronio
2010 arXiv   pre-print
This survey article describes the algorithmic approaches successfully used over the time to construct hyperbolic structures on 3-dimensional topological "objects" of various types, and to classify several  ...  classes of such objects using such structures.  ...  Hyperbolic ideal tetrahedra Let us start from a compact 3-manifold M with non-empty boundary consisting of tori, and from an ideal triangulation T of M.  ... 
arXiv:1003.4739v1 fatcat:7j3aci52sjchjheeke3cyqkkve

Minimum volume hyperbolic 3-manifolds

Peter Milley
2009 Journal of Topology  
In so doing we complete the proof that the Weeks manifold is the minimum-volume compact hyperbolic 3-manifold, as well as enumerating the 10 smallest one-cusped hyperbolic 3-manifolds.  ...  We enumerate the small-volume manifolds that can be obtained by Dehn filling on Mom-2 and Mom-3 manifolds as defined by Gabai, Meyerhoff, and the author.  ...  I am also interested in alternate ways of defining hyperbolic structures on 3-manifolds, other than with ideal triangulations.  ... 
doi:10.1112/jtopol/jtp006 fatcat:oetweglkcng7ljcco4x62rf3fa

A Census of Tetrahedral Hyperbolic Manifolds

Evgeny Fominykh, Stavros Garoufalidis, Matthias Goerner, Vladimir Tarkaev, Andrei Vesnin
2016 Experimental Mathematics  
We call a cusped hyperbolic 3-manifold tetrahedral if it can be decomposed into regular ideal tetrahedra.  ...  Following an earlier publication by three of the authors, we give a census of all tetrahedral manifolds and all of their combinatorial tetrahedral tessellations with at most 25 (orientable case) and 21  ...  ., V.T. and A.V. were supported in part by the Ministry of Education and Science of the Russia (the state task number 1.1260.2014/K) and RFBR grant 16-01-00414.  ... 
doi:10.1080/10586458.2015.1114436 fatcat:qpsgwhkemjbnbe75jbjoky74ii

A census of tetrahedral hyperbolic manifolds [article]

Evgeny Fominykh, Stavros Garoufalidis, Matthias Goerner, Vladimir Tarkaev, Andrei Vesnin
2015 arXiv   pre-print
We call a cusped hyperbolic 3-manifold tetrahedral if it can be decomposed into regular ideal tetrahedra.  ...  Following an earlier publication by three of the authors, we give a census of all tetrahedral manifolds and all of their combinatorial tetrahedral tessellations with at most 25 (orientable case) and 21  ...  ., V.T. and A.V. were supported in part by the Ministry of Education and Science of the Russia (the state task number 1.1260.2014/K) and RFBR grant 16-01-00414.  ... 
arXiv:1502.00383v2 fatcat:5xqu4x65t5d7jelcm3gh2altju

Many cusped hyperbolic 3-manifolds do not bound geometrically [article]

Alexander Kolpakov, Alan W. Reid, Stefano Riolo
2019 arXiv   pre-print
In this note, we show that there exist cusped hyperbolic 3-manifolds that embed geodesically, but cannot bound geometrically.  ...  Our result complements the work by Long and Reid on geometric boundaries of compact hyperbolic 4-manifolds, and by Kolpakov, Reid and Slavich on embedding arithmetic hyperbolic manifolds.  ...  Proof of Theorem 1.5: We will adopt the usual ideal triangulations of the figure-eight knot complement and its sibling manifold by regular ideal hyperbolic tetrahedra.  ... 
arXiv:1811.05509v3 fatcat:qnpp4hwkgfbrfmayjquvjhhw6a

Algebraic and geometric solutions of hyperbolicity equations

Stefano Francaviglia
2004 Topology and its Applications  
In this paper, we study the differences between algebraic and geometric solutions of hyperbolicity equations for ideally triangulated 3-manifolds, and their relations with the variety of representations  ...  In the last section we study some examples, doing explicit calculations for three interesting manifolds. Proposition 4. Let M be an ideally triangulated manifold and let z be a choice of moduli.  ...  Ideal triangulations with moduli We fix here the class of manifolds we consider, namely the class of ideally triangulated cusped 3-manifolds.  ... 
doi:10.1016/j.topol.2004.06.005 fatcat:oy2qlabz7vbtrkk2fqiqznftpy

Computing complete hyperbolic structures on cusped 3-manifolds [article]

Clément Maria, Owen Rouillé
2021 arXiv   pre-print
We focus on the computation of a complete hyperbolic structure on a connected orientable hyperbolic 3-manifold with torus boundaries. This family of 3-manifolds includes the knot complements.  ...  A fundamental way to study 3-manifolds is through the geometric lens, one of the most prominent geometries being the hyperbolic one.  ...  Let T be an ideal triangulation of M , a non-compact orientable 3-manifold with toric cusps.  ... 
arXiv:2112.06360v1 fatcat:qa7wm7d6xvam3oxoahozchbsha

Triangulations of 3-manifolds, hyperbolic relative handlebodies, and Dehn filling

François Costantino, Roberto Frigerio, Bruno Martelli, Carlo Petronio
2007 Commentarii Mathematici Helvetici  
As consequences of our constructions, we also show that: • A triangulation of a 3-manifold is uniquely determined up to isotopy by its 1-skeleton; • If a 3-manifold M has an ideal triangulation with edges  ...  volume of a hyperbolic regular ideal octahedron.  ...  We thank Simon King for communicating to us a proof, based on traditional cut-and-paste techniques, of Theorem 1.2 for the special case of genuine triangulations (without multiple and self-adjacencies)  ... 
doi:10.4171/cmh/114 fatcat:aodsonqpj5g6bp6lg6pjr22bcu

Triangulations of 3-manifolds, hyperbolic relative handlebodies, and Dehn filling [article]

Francois Costantino, Roberto Frigerio, Bruno Martelli, Carlo Petronio
2005 arXiv   pre-print
As consequences of our constructions, we also show that: - A triangulation of a 3-manifold is uniquely determined up to isotopy by its 1-skeleton; - If a 3-manifold M has an ideal triangulation with edges  ...  of valence at least 6, then M is hyperbolic and the edges are homotopically non-trivial, whence homotopic to geodesics; - Every finite group G is the isometry group of a closed hyperbolic 3-manifold with  ...  Acknowledgments We thank Simon King for communicating to us a proof, based on traditional cut-and-paste techniques, of Theorem 0.2 for the special case of genuine triangulations (without multiple and self-adjacencies  ... 
arXiv:math/0402339v2 fatcat:37fxquveofarxdlhzsndoqvfni

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

Paola Cristofori, Evgeny Fominykh, Michele Mulazzani, Vladimir Tarkaev
2017 RACSAM  
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  ...  Moreover, in Section 3 we give two sided bounds for the graph complexity of compact tetrahedral manifolds (i.e., manifolds admitting a triangulation by regular ideal hyperbolic tetrahedra).  ... 
doi:10.1007/s13398-017-0478-4 fatcat:fdlnzmqigbcz5bavwj7hdatl6y
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