Connecting essential triangulations II: via 2-3 moves only

TL;DR

This paper demonstrates that the set of essential ideal triangulations of a manifold is connected through specific moves, particularly 2-3 and 3-2 moves, under certain conditions, simplifying the understanding of their connectivity.

Contribution

It proves that essential ideal triangulations are connected via 2-3 and 3-2 moves alone when excluding certain non-preserving moves, and re-establishes full connectivity with V-moves included.

Findings

Essential ideal triangulations are connected via 2-3 and 3-2 moves under specific conditions.

Full connectivity is restored when V-moves are included.

Results also apply to L-essential triangulations.

Abstract

In previous work we showed that for a manifold $M$, whose universal cover has infinitely many boundary components, the set of essential ideal triangulations of $M$ is connected via 2-3, 3-2, 0-2, and 2-0 moves. Here we show that this set is also connected via 2-3 and 3-2 moves alone, if we ignore those triangulations for which no 2-3 move preserves essentiality. If we also allow V-moves and their inverses then the full set of essential ideal triangulations of $M$ is once again connected. These results also hold if we replace essential triangulations with $L$-essential triangulations.