Connecting 3-manifold triangulations with unimodal sequences of elementary moves

TL;DR

This paper introduces the concept of unimodal sequences of elementary moves connecting 3-manifold triangulations, proving their existence for certain cases and exploring their practical utility through algorithms and experiments.

Contribution

It establishes the existence of unimodal sequences of Pachner moves between triangulations and develops algorithms to find such sequences, enhancing understanding of their structure.

Findings

Unimodal sequences exist between certain 3-manifold triangulations.

Algorithms can effectively find unimodal sequences in practice.

Computational experiments demonstrate the utility of unimodal sequences.

Abstract

A key result in computational 3-manifold topology is that any two triangulations of the same 3-manifold are connected by a finite sequence of bistellar flips, also known as Pachner moves. One limitation of this result is that little is known about the structure of this sequence; knowing more about the structure could help both proofs and algorithms. Motivated by this, we consider sequences of moves that are "unimodal" in the sense that they break up into two parts: first, a sequence that monotonically increases the size of the triangulation; and second, a sequence that monotonically decreases the size. We prove that any two one-vertex triangulations of the same 3-manifold, each with at least two tetrahedra, are connected by a unimodal sequence of 2-3 and 2-0 moves. We also study the practical utility of unimodal sequences; specifically, we implement an algorithm to find such sequences, and use this algorithm to perform some detailed computational experiments.