## Geometry & Topology

### $1$–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.

#### Article information

Source
Geom. Topol., Volume 19, Number 5 (2015), 2619-2689.

Dates
Accepted: 9 January 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510858846

Digital Object Identifier
doi:10.2140/gt.2015.19.2619

Mathematical Reviews number (MathSciNet)
MR3416111

Zentralblatt MATH identifier
1330.57029

#### Citation

Garoufalidis, Stavros; Hodgson, Craig D; Rubinstein, J Hyam; Segerman, Henry. $1$–efficient triangulations and the index of a cusped hyperbolic $3$–manifold. Geom. Topol. 19 (2015), no. 5, 2619--2689. doi:10.2140/gt.2015.19.2619. https://projecteuclid.org/euclid.gt/1510858846

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