Abstract | ||
---|---|---|
A triangulation of a $3$-manifold can be shown to be homeomorphic to the $3$-sphere by describing a discrete Morse function on it with only two critical faces, that is, a sequence of elementary collapses from the triangulation with one tetrahedron removed down to a single vertex. Unfortunately, deciding whether such a sequence exist is believed to be very difficult in general. this article we present a method, based on uniform spanning trees, to estimate how difficult it is to collapse a given $3$-sphere triangulation after removing a tetrahedron. In addition we show that out of all $3$-sphere triangulations with eight vertices or less, exactly $22$ admit a non-collapsing sequence onto a contractible non-collapsible $2$-complex. As a side product we classify all minimal triangulations of the dunce hat, and all contractible non-collapsible $2$-complexes with at most $18$ triangles. This is complemented by large scale experiments on the collapsing difficulty of $9$- and $10$-vertex spheres. Finally, we propose an easy-to-compute characterisation of $3$-sphere triangulations which experimentally exhibit a low proportion of collapsing sequences, leading to a heuristic to produce $3$-sphere triangulations with difficult combinatorial properties. |
Year | Venue | Field |
---|---|---|
2015 | arXiv: Geometric Topology | Topology,Combinatorics,Vertex (geometry),Contractible space,Triangulation (social science),Spanning tree,Tetrahedron,Dunce hat,Mathematics,Morse theory,Point set triangulation |
DocType | Volume | Citations |
Journal | abs/1509.07607 | 0 |
PageRank | References | Authors |
0.34 | 3 | 2 |
Name | Order | Citations | PageRank |
---|---|---|---|
Joao Paixao | 1 | 7 | 3.57 |
Jonathan Spreer | 2 | 47 | 11.46 |