Name
Abstract | ||
|---|---|---|
Discrete Morse theory emerged as an essential tool for computational geometry and topology. Its core structures are discrete gradient fields, defined as acyclic matchings on a complex $C$, from which topological and geometrical informations of $C$ can be efficiently computed, in particular its homology or Morse-Smale decomposition. Given a function $f$ sampled on $C$, it is possible to derive a discrete gradient that mimics the dynamics of $f$. Many such constructions are based on some variant of a greedy pairing of adjacent cells, given an appropriate weighting. However, proving that the dynamics of $f$ is correctly captured by this process is usually intricate. This work introduces the notion of discrete smoothness of the pair $(f,C)$, as a minimal sampling condition to ensure that the discrete gradient is geometrically faithful to $f$. More precisely, a discrete gradient construction from a function $f$ on a polyhedron complex $C$ of any dimension is studied, leading to theoretical guarantees prior to the discrete smoothness assumption. Those results are then extended and completed for the smooth case. As an application, a purely combinatorial proof that all CAT(0) cube complexes are collapsible is given. |
Year | Venue | Field |
|---|---|---|
2018 | arXiv: Geometric Topology | Combinatorics,Weighting,Polyhedron,Computational geometry,Pairing,Combinatorial proof,Discrete Morse theory,Smoothness,Mathematics,Cube |
DocType | Volume | Citations |
Journal | abs/1801.10118 | 0 |
PageRank | References | Authors |
0.34 | 2 | 4 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Joao Paixao | 1 | 7 | 3.57 |
| Joao Lagoas | 2 | 0 | 0.34 |
| Thomas Lewiner | 3 | 700 | 43.70 |
| Tiago Novello | 4 | 0 | 0.68 |