Braids, complex volume and cluster algebras (Q498732)

The purpose of this article is to give a cluster-algebraic interpretation of the complex volume of knots. In [\textit{K. Hikami} and \textit{R. Inoue}, J. Knot Theory Ramifications 23, No. 1, Article ID 1450006, 33 p. (2014; Zbl 1314.57005)] the authors gave an interpretation of the cluster mutation as a hyperbolic ideal tetrahedron, and proposed a method to compute the complex volume of 2-bridge knots. In this article, the authors give a geometric interpretation of the R-operator via a hyperbolic octahedron composed of four ideal tetrahedra, the cluster variable corresponding to the edge parameter used by \textit{C. K. Zickert} [Duke Math. J. 150, No. 3, 489--532 (2009; Zbl 1246.58019)]. The main result of the paper is a formula, in the spirit of the method of Zickert, for computing complex volume in terms of cluster variables.