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

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.

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
Received: 16 July 2018
Revised: 7 February 2019
Accepted: 20 February 2019
Published: 23 February 2020
Jinseok Cho
Department of Mathematics Education
Busan National University of Education
South Korea
Seokbeom Yoon
School of Mathematics
Korea Institute for Advanced Study
South Korea
Christian K Zickert
Department of Mathematics
University of Maryland
College Park, MD
United States