On the Hikami-Inoue conjecture

Abstract

Given a braid presentation \(\sigma\) of a hyperbolic knot \(K\), Hikami and Inoue consider a system of polynomial equations arising from a sequence of cluster mutations determined by \(\sigma\). They show that any solution gives rise to shape variables and thus determines a boundary-parabolic \(\mathrm{PSL}(2,\mathbb{C})\)-representation of \(\pi_1(S^3\setminus K)\). They conjecture the existence of a solution corresponding to the geometric representation. We show that a boundary-parabolic representation \(\rho\) arises from a solution if and only if the length of \(\sigma\) modulo \(2\) equals the obstruction to lifting \(\rho\) to a boundary-parabolic \(\mathrm{SL}(2,\mathbb{C})\)-representation (an element in \(\mathbb{Z}_2\)). In particular, the Hikami-Inoue conjecture holds if and only if the length of \(\sigma\) is odd. This can always be achieved by adding a kink to the braid if necessary. We also explicitly construct the solution corresponding to a boundary-parabolic representation given in the Wirtinger presentation of \(\pi_1(S^3\setminus K)\).