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On the Hikami–Inoue
conjecture
Jinseok Cho, Seokbeom Yoon and Christian K Zickert |
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Algebraic & Geometric Topology 20 (2020) 279–301
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Abstract |
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Given a braid presentation of a hyperbolic knot, Hikami and Inoue consider a system of polynomial equations arising from a sequence of cluster mutations determined by . They show that any solution gives rise to shape parameters and thus determines a boundary-parabolic –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 arises from a solution if and only if the length of modulo equals the obstruction to lifting to a boundary-parabolic –representation (as an element in ). In particular, the Hikami–Inoue conjecture holds if and only if the length of 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
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Mathematical Subject Classification 2010
Primary: 57M05, 57M25, 57M50, 57M60, 57N16
Secondary: 13F60
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References |
Publication
Received: 16 July 2018
Revised: 7 February 2019
Accepted: 20 February 2019
Published: 23 February 2020
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Authors |
| Jinseok Cho | |
| Department of Mathematics
Education Busan National University of Education Busan South Korea |
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| Seokbeom Yoon | |
| School of Mathematics Korea Institute for Advanced Study Seoul South Korea |
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| Christian K Zickert | |
| Department of Mathematics University of Maryland College Park, MD United States |
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