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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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 23 and 02 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
Received: 30 October 2013
Accepted: 9 January 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
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

Subjects
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}

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

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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