Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces

TL;DR

This paper establishes bounds and algorithms for topological simplification of triangulated surfaces, linking discrete and Riemannian systolic inequalities, and analyzing decompositions and embeddings of surfaces.

Contribution

It translates Riemannian systolic inequalities into a discrete setting, proves new bounds for surface decompositions, and analyzes embeddings of cut graphs with given combinatorial maps.

Findings

Proves a conjecture by Przytycka and Przytycki from 1993.

Provides an O(gn)-time algorithm for pants decompositions.

Shows that certain embeddings can have superlinear length in the number of triangles.

Abstract

How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (or their dual cross-metric counterpart). Our work builds upon Riemannian systolic inequalities, which bound the minimum length of non-trivial closed curves in terms of the genus and the area of the surface. We first describe a systematic way to translate Riemannian systolic inequalities to a discrete setting, and vice-versa. This implies a conjecture by Przytycka and Przytycki from 1993, a number of new systolic inequalities in the discrete setting, and the fact that a theorem of Hutchinson on the edge-width of triangulated surfaces and Gromov's systolic inequality for surfaces are essentially equivalent. We also discuss how these proofs generalize to higher dimensions. Then we focus on topological decompositions of surfaces. Relying on ideas of Buser, we prove the existence of pants decompositions of length O(g^{3/2}n^{1/2}) for any triangulated combinatorial surface of genus g with n triangles, and describe an O(gn)-time algorithm to compute such a decomposition. Finally, we consider the problem of embedding a cut graph (or more generally a cellular graph) with a given combinatorial map on a given surface. Using random triangulations, we prove (essentially) that, for any choice of a combinatorial map, there are some surfaces on which any cellular embedding with that combinatorial map has length superlinear in the number of triangles of the triangulated combinatorial surface. There is also a similar result for graphs embedded on polyhedral triangulations.