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

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

doi:10.2140/gt.2015.19.2619

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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

DOI: 10.2140/gt.2015.19.2619

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.

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/

Volume 19, issue 5 (2015)