#### Volume 20, issue 1 (2020)

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On the Hikami–Inoue conjecture

### Jinseok Cho, Seokbeom Yoon and Christian K Zickert

Algebraic & Geometric Topology 20 (2020) 279–301
##### Abstract

Given a braid presentation $D$ of a hyperbolic knot, Hikami and Inoue consider a system of polynomial equations arising from a sequence of cluster mutations determined by $D\phantom{\rule{-0.17em}{0ex}}$. They show that any solution gives rise to shape parameters and thus determines a boundary-parabolic $PSL\left(2,ℂ\right)$–representation of the knot group. They conjecture the existence of a solution corresponding to the geometric representation. Here we 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 $SL\left(2,ℂ\right)$–representation (as an element in ${ℤ}_{2}$). In particular, the Hikami–Inoue conjecture holds if and only if the length of $D$ 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 the knot group.

##### Keywords
Hikami–Inoue conjecture, Ptolemy variety, braid, hyperbolic knot, boundary-parabolic representation, cluster coordinates
##### Mathematical Subject Classification 2010
Primary: 57M05, 57M25, 57M50, 57M60, 57N16
Secondary: 13F60