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. They show that any solution gives rise to shape parameters and thus determines a boundary-parabolic PSL(2, )–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, )–representation (as an element in 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
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