Braids, complex volume and cluster algebras

Hikami, Kazuhiro; Inoue, Rei

doi:10.2140/agt.2015.15.2175

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Braids, complex volume and cluster algebras

Kazuhiro Hikami and Rei Inoue

Algebraic & Geometric Topology 15 (2015) 2175–2194

DOI: 10.2140/agt.2015.15.2175

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

Volume 15, issue 4 (2015)