On the Hikami–Inoue conjecture

Cho, Jinseok; Yoon, Seokbeom; Zickert, Christian

doi:10.2140/agt.2020.20.279

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On the Hikami–Inoue conjecture

Jinseok Cho, Seokbeom Yoon and Christian K Zickert

Algebraic & Geometric Topology 20 (2020) 279–301

DOI: 10.2140/agt.2020.20.279

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 $PSL (2, ℂ)$ –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 $ρ$ arises from a solution if and only if the length of $D$ modulo $2$ equals the obstruction to lifting $ρ$ to a boundary-parabolic $SL (2, ℂ)$ –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.

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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

References

Publication

Received: 16 July 2018

Revised: 7 February 2019

Accepted: 20 February 2019

Published: 23 February 2020

Authors

Jinseok Cho
Department of Mathematics Education Busan National University of Education Busan South Korea

Seokbeom Yoon
School of Mathematics Korea Institute for Advanced Study Seoul South Korea

Christian K Zickert
Department of Mathematics University of Maryland College Park, MD United States

Volume 20, issue 1 (2020)