License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2023.21
URN: urn:nbn:de:0030-drops-178712
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/17871/
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Burton, Benjamin A. ; He, Alexander

Finding Large Counterexamples by Selectively Exploring the Pachner Graph

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LIPIcs-SoCG-2023-21.pdf (0.8 MB)

Abstract

We often rely on censuses of triangulations to guide our intuition in 3-manifold topology. However, this can lead to misplaced faith in conjectures if the smallest counterexamples are too large to appear in our census. Since the number of triangulations increases super-exponentially with size, there is no way to expand a census beyond relatively small triangulations - the current census only goes up to 10 tetrahedra. Here, we show that it is feasible to search for large and hard-to-find counterexamples by using heuristics to selectively (rather than exhaustively) enumerate triangulations. We use this idea to find counterexamples to three conjectures which ask, for certain 3-manifolds, whether one-vertex triangulations always have a "distinctive" edge that would allow us to recognise the 3-manifold.

BibTeX - Entry

@InProceedings{burton_et_al:LIPIcs.SoCG.2023.21,
  author =	{Burton, Benjamin A. and He, Alexander},
  title =	{{Finding Large Counterexamples by Selectively Exploring the Pachner Graph}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{21:1--21:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/17871},
  URN =		{urn:nbn:de:0030-drops-178712},
  doi =		{10.4230/LIPIcs.SoCG.2023.21},
  annote =	{Keywords: Computational topology, 3-manifolds, Triangulations, Counterexamples, Heuristics, Implementation, Pachner moves, Bistellar flips}
}

Keywords: Computational topology, 3-manifolds, Triangulations, Counterexamples, Heuristics, Implementation, Pachner moves, Bistellar flips
Collection: 39th International Symposium on Computational Geometry (SoCG 2023)
Issue Date: 2023
Date of publication: 09.06.2023
Supplementary Material: Software (Source Code): https://github.com/AlexHe98/triang-counterex

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