Finding large counterexamples by selectively exploring the Pachner graph
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917606315786240 |
|---|---|
| author | Burton, Benjamin A. He, Alexander |
| author_facet | Burton, Benjamin A. He, Alexander |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_06321 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Finding large counterexamples by selectively exploring the Pachner graph Burton, Benjamin A. He, Alexander Geometric Topology Computational Geometry 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. |
| title | Finding large counterexamples by selectively exploring the Pachner graph |
| topic | Geometric Topology Computational Geometry |
| url | https://arxiv.org/abs/2303.06321 |