1-efficient triangulations and the index of a cusped hyperbolic 3-manifold

TL;DR

This paper demonstrates that the 3D index of an ideal triangulation can serve as a topological invariant for cusped hyperbolic 3-manifolds, linking index structures to 1-efficiency and move invariance.

Contribution

It establishes the equivalence between index structures and 1-efficiency and shows the invariance of the 3D index under certain moves for hyperbolic manifolds.

Findings

Index structure exists if and only if the triangulation is 1-efficient

The 3D index is invariant under 2-3 and 0-2 moves for hyperbolic manifolds

Canonical sets of 1-efficient triangulations are related by these moves

Abstract

In this paper we will promote the 3D index of an ideal triangulation T of an oriented cusped 3-manifold M (a collection of q-series with integer coefficients, introduced by Dimofte-Gaiotto-Gukov) to a topological invariant of oriented cusped hyperbolic 3-manifolds. To achieve our goal we show that (a) T admits an index structure if and only if T is 1-efficient and (b) if M is hyperbolic, it has a canonical set of 1-efficient ideal triangulations related by 2-3 and 0-2 moves which preserve the 3D index. We illustrate our results with several examples.