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Braids, complex volume and cluster algebras

Kazuhiro Hikami and Rei Inoue

Algebraic & Geometric Topology 15 (2015) 2175–2194
Abstract

We try to give a cluster-algebraic interpretation of the complex volume of knots. We construct the R–operator from cluster mutations, and show that it can be regarded as a hyperbolic octahedron. The cluster variables are interpreted as the edge parameters used by Zickert for computing complex volume.

Keywords
knot, hyperbolic volume, complex volume, cluster algebra
Mathematical Subject Classification 2010
Primary: 57M25
Secondary: 13F60
References
Publication
Received: 28 April 2014
Revised: 6 November 2014
Accepted: 14 November 2014
Published: 10 September 2015
Authors
Kazuhiro Hikami
Faculty of Mathematics
Kyushu University
Fukuoka 819-0395
Japan
Rei Inoue
Department of Mathematics and Informatics, Faculty of Science
Chiba University
Chiba 263-8522
Japan