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 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) 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.