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

Abstract

In this paper we will promote the 3D index of an ideal triangulation $\mathcal{T}$ of an oriented cusped $3$–manifold $M$ (a collection of $q$–series with integer coefficients, introduced by Dimofte, Gaiotto and Gukov) to a topological invariant of oriented cusped hyperbolic $3$–manifolds. To achieve our goal we show that (a) $\mathcal{T}$ admits an index structure if and only if $\mathcal{T}$ 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.