License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2016.64
URN: urn:nbn:de:0030-drops-64050
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6405/
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Maria, Clément ; Spreer, Jonathan

Admissible Colourings of 3-Manifold Triangulations for Turaev-Viro Type Invariants

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Abstract

Turaev-Viro invariants are amongst the most powerful tools to distinguish 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them rely on the enumeration of an extremely large set of combinatorial data defined on the triangulation, regardless of the underlying topology of the manifold.

In the article, we propose a finer study of these combinatorial data, called admissible colourings, in relation with the cohomology of the manifold. We prove that the set of admissible colourings to be considered is substantially smaller than previously known, by furnishing new upper bounds on its size that are aware of the topology of the manifold. Moreover, we deduce new topology-sensitive enumeration algorithms based on these bounds.

The paper provides a theoretical analysis, as well as a detailed experimental study of the approach. We give strong experimental evidence on large manifold censuses that our upper bounds are tighter than the previously known ones, and that our algorithms outperform significantly state of the art implementations to compute Turaev-Viro invariants.

BibTeX - Entry

@InProceedings{maria_et_al:LIPIcs:2016:6405,
  author =	{Cl{\'e}ment Maria and Jonathan Spreer},
  title =	{{Admissible Colourings of 3-Manifold Triangulations for Turaev-Viro Type Invariants}},
  booktitle =	{24th Annual European Symposium on Algorithms (ESA 2016)},
  pages =	{64:1--64:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-015-6},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{57},
  editor =	{Piotr Sankowski and Christos Zaroliagis},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6405},
  URN =		{urn:nbn:de:0030-drops-64050},
  doi =		{10.4230/LIPIcs.ESA.2016.64},
  annote =	{Keywords: low-dimensional topology, triangulations of 3-manifolds,  cohomology theory, Turaev-Viro invariants,  combinatorial algorithms}
}

Keywords: low-dimensional topology, triangulations of 3-manifolds, cohomology theory, Turaev-Viro invariants, combinatorial algorithms
Collection: 24th Annual European Symposium on Algorithms (ESA 2016)
Issue Date: 2016
Date of publication: 18.08.2016

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