Connecting essential triangulations I: via 2-3 and 0-2 moves

TL;DR

This paper proves the existence and connectivity of essential ideal triangulations for certain 3-manifolds, introduces L-essential triangulations, and applies these results to representations and invariants in 3-manifold topology.

Contribution

It establishes conditions for essential triangulations to exist and be connected via specific moves, introduces L-essential triangulations, and links these to Thurston's equations and quantum invariants.

Findings

Essential triangulations exist under certain conditions on the universal cover.

Such triangulations are connected via 2-3, 3-2, 0-2, and 2-0 moves.

The invariance of the 1-loop invariant is proven for these triangulations.

Abstract

Suppose that $M$ is a compact, connected three-manifold with boundary. We show that if the universal cover has infinitely many boundary components then $M$ has an ideal triangulation which is essential: no edge can be homotoped into the boundary. Under the same hypotheses, we show that the set of essential triangulations of $M$ is connected via 2-3, 3-2, 0-2, and 2-0 moves. The above results are special cases of our general theory. We introduce $L$-essential triangulations: boundary components of the universal cover receive labels and no edge has the same label at both ends. As an application, under mild conditions on a representation, we construct an ideal triangulation for which a solution to Thurston's gluing equations recovers the given representation. Our results also imply that such triangulations are connected via 2-3, 3-2, 0-2, and 2-0 moves. Together with results of Pandey and Wong, this proves that Dimofte and Garoufalidis' 1-loop invariant is independent of the choice of essential triangulation.