Geometry & Topology

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

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

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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 23 and 02 moves which preserve the 3D index. We illustrate our results with several examples.

Article information

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

Received: 30 October 2013
Accepted: 9 January 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx] 57M50: Geometric structures on low-dimensional manifolds
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

ideal triangulations hyperbolic $3$–manifolds gluing equations 3D index invariants $1$–efficient triangulations


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.

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