On the Hikami-Inoue conjecture

TL;DR

This paper proves the Hikami-Inoue conjecture by linking solutions of polynomial equations from braid presentations to boundary-parabolic representations, showing the conjecture holds when the braid length is odd, and provides explicit constructions.

Contribution

It establishes a precise criterion connecting braid length parity to the existence of solutions corresponding to boundary-parabolic representations and offers explicit solution constructions.

Findings

The Hikami-Inoue conjecture holds if and only if the braid length is odd.

A boundary-parabolic representation arises from a solution iff the braid length mod 2 matches the lifting obstruction.

Explicit solutions can be constructed from the Wirtinger presentation of the knot group.

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 $D$. They show that any solution gives rise to shape parameters and thus determines a boundary-parabolic $\mathrm{PSL}(2,\mathbb{C})$-representation of the knot group. They conjecture the existence of a solution corresponding to the geometric representation. In this paper, 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 $\mathrm{SL}(2,\mathbb{C})$-representation (as an element in $\mathbb{Z}_2$). 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.