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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 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.

Keywords
ideal triangulations, hyperbolic $3$–manifolds, gluing equations 3D index, invariants, $1$–efficient triangulations
Mathematical Subject Classification 2010
Primary: 57N10, 57M50
Secondary: 57M25
References
Publication
Received: 30 October 2013
Accepted: 9 January 2015
Published: 20 October 2015
Proposed: Cameron Gordon
Seconded: Peter Teichner, Robion Kirby
Authors
Stavros Garoufalidis
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30332-0160
USA
http://www.math.gatech.edu/~stavros
Craig D Hodgson
Department of Mathematics and Statistics
The University of Melbourne
Melbourne, Parkville VIC 3010
Australia
http://www.ms.unimelb.edu.au/~cdh/
J Hyam Rubinstein
Department of Mathematics and Statistics
The University of Melbourne
Melbourne, Parkville VIC 3010
Australia
http://www.ms.unimelb.edu.au/~rubin
Henry Segerman
Department of Mathematics
Oklahoma State University
Stillwater, VIC 74078
USA
Department of Mathematics and Statistics
The University of Melbourne
Melbourne, Parkville VIC 3010
Australia
http://math.okstate.edu/people/segerman/