Connectivity of triangulations without degree one edges under 2-3 and 3-2 moves

TL;DR

This paper extends known connectivity results of the Pachner graph to subgraphs of triangulations without degree one edges, demonstrating their connectivity under 2-3 and 3-2 moves for specific classes of 3-manifolds.

Contribution

It proves that the subgraph of triangulations without degree one edges remains connected under 2-3 and 3-2 moves, with few exceptions, for certain 3-manifold classes.

Findings

Subgraph of triangulations without degree one edges is connected.

Connectivity holds for single-vertex closed manifold triangulations.

Connectivity holds for ideal triangulations with non-spherical boundary components.

Abstract

Matveev and Piergallini independently showed that, with a small number of known exceptions, any triangulation of a three-manifold can be transformed into any other triangulation of the same three-manifold with the same number of vertices, via a sequence of 2-3 and 3-2 moves. We can interpret this as showing that the Pachner graph of such triangulations is connected. In this paper, we extend this result to show that (again with a small number of known exceptions), the subgraph of the Pachner graph consisting of triangulations without degree one edges is also connected, for single-vertex triangulations of closed manifolds, and ideal triangulations of manifolds with non-spherical boundary components.