Connecting 3-manifold triangulations with monotonic sequences of bistellar flips

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 sequences; knowing more about the structure could help both proofs and algorithms. Motivated by this, we show that there must be a sequence that satisfies a rigid property that we call "semi-monotonicity". We also study this result empirically: we implement an algorithm to find such semi-monotonic sequences, and compare their characteristics to less structured sequences, in order to better understand the practical and theoretical utility of this result.