🤖 AI Summary
This paper addresses the Pachner-move connectivity problem for one-vertex triangulations of closed 3-manifolds: given any two homeomorphic triangulations each containing at least two tetrahedra, does there always exist a “unimodal” sequence—first monotonically increasing, then monotonically decreasing, in tetrahedron count—that connects them? Through combinatorial topology and computational 3-manifold theory, we provide the first rigorous proof of this unimodal connectivity theorem. We further design and implement a custom algorithm to conduct large-scale numerical experiments, empirically verifying feasibility and characterizing typical sequence lengths. Our results establish a fundamental structural constraint for 3-manifold classification, yield new tools for normal surface algorithms, and advance automated theorem proving in low-dimensional topology.
📝 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.