Volume 20, issue 1 (2020)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 24
Issue 2, 595–1223
Issue 1, 1–594

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
Other MSP Journals
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