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.
@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/entities/document/10.4230/LIPIcs.SoCG.2023.21}, 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} }
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