Connecting 3-manifold triangulations with unimodal sequences of elementary moves

TL;DR

The paper addresses how to connect two one-vertex triangulations of a closed $3$-manifold with at least two tetrahedra using structured unimodal sequences of moves. It develops walls and arches, adapting Matveev’s arch-with-membrane idea to convert non-unimodal progress into a unimodal path built from moves $2$-$3$ and $2$-$0$, leveraging the duality with special spines. The main theoretical contribution is Theorem semiMono, proving the existence of unimodal $2$-$3$/$2$-$0$ sequences between any such triangulations (extendable to broader settings), along with discussion toward a fuller unimodal $2$-$3$/$3$-$2$ conjecture. Complementarily, the authors implement algorithms to search for unimodal sequences and compare their performance to blind searches, finding that unimodal paths are feasible and that $2$-$0$ moves are typically not needed in practice, informing both proofs and computations in the homeomorphism problem.

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.

Figures (16)

• Figure 1: The 2-3 and 3-2 moves.
• Figure 2: The 1-4 and 4-1 moves.
• Figure 3: The 0-2 and 2-0 moves.
• Figure 4: A fragment of the link of $v$, built from triangles.
• Figure 5: Converting a triangulation into its dual spine.

Theorems & Definitions (11)

• Remark 1
• definition 2
• definition 3
• definition 4
• Theorem 5: Matveev Matveev2007 and Piergallini Piergallini1988, independently
• Theorem 7: Amendola Amendola2005
• Conjecture 7
• Theorem 8
• Lemma 10
• Theorem 10