#### Volume 19, issue 5 (2015)

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

### Stavros Garoufalidis, Craig D Hodgson, J Hyam Rubinstein and Henry Segerman

Geometry & Topology 19 (2015) 2619–2689
##### Abstract

In this paper we will promote the 3D index of an ideal triangulation $\mathsc{T}$ 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) $\mathsc{T}$ admits an index structure if and only if $\mathsc{T}$ 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.

##### Keywords
ideal triangulations, hyperbolic $3$–manifolds, gluing equations 3D index, invariants, $1$–efficient triangulations
##### Mathematical Subject Classification 2010
Primary: 57N10, 57M50
Secondary: 57M25