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 "monotonic" 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 monotonic sequence of 2-3 and 2-0 moves. We also study the practical utility of monotonic sequences; specifically, we implement an algorithm to find such sequences, and use this algorithm to perform some detailed computational experiments.

翻译：在计算三维流形拓扑学中，一个关键结论是：同一三维流形的任意两个三角剖分均可通过有限序列的双星翻转（亦称Pachner移动）相互转换。该结论的一个局限在于，人们对此序列的结构知之甚少；而更深入地理解其结构将有助于理论证明与算法设计。受此启发，我们研究具有“单调性”的移动序列，其可分解为两部分：首先是单调增加三角剖分规模的子序列，随后是单调减小规模的子序列。我们证明：对于同一三维流形的任意两个单顶点三角剖分（每个剖分至少包含两个四面体），均存在由2-3移动和2-0移动构成的单调序列将其连通。同时，我们探究了单调序列的实际效用：具体而言，我们实现了寻找此类序列的算法，并利用该算法开展了若干精细的计算实验。