Cluster variables on a braid

Given a braid presentation $D$ of a hyperbolic knot, Hikami and Inoue considered equations arising from a sequence of cluster mutations determined by $D$. They showed that any solution of these equations determines a boundary-parabolic $PSL(2;\mathbb{C})$-representation of the knot group. They also conjectured the existence of solution corresponding to the geometric representation. In this talk we will show that a boundary-parabolic representation $\rho$ arises from a solution if and only if the length of $D$ modulo 2 equals the obstruction to lifting $\rho$ to a boundary-parabolic $PSL(2;\mathbb{C})$-representation. In particular, the Hikami-Inoue conjecture holds if and only if the length of $D$ is odd. This work is joint with Jinseok Cho and Christian Zickert.