Geodesic
On the Hikami–Inoue conjecture
Cho, Jinseok1 ; Yoon, Seokbeom2 ; Zickert, Christian3
1Department of Mathematics Education, Busan National University of Education, Busan, South Korea
2School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea
3Department of Mathematics, University of Maryland, College Park, MD, United States
Algebraic and Geometric Topology, Tome 20 (2020) no. 1, pp. 279-301
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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 ℤ2). 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.

DOI : 10.2140/agt.2020.20.279
Classification : 57M05, 57M25, 57M50, 57M60, 57N16, 13F60
Keywords: Hikami–Inoue conjecture, Ptolemy variety, braid, hyperbolic knot, boundary-parabolic representation, cluster coordinates

Cho, Jinseok  1   ; Yoon, Seokbeom  2   ; Zickert, Christian  3

1 Department of Mathematics Education, Busan National University of Education, Busan, South Korea
2 School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea
3 Department of Mathematics, University of Maryland, College Park, MD, United States 
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Cho, Jinseok; Yoon, Seokbeom; Zickert, Christian. On the Hikami–Inoue conjecture. Algebraic and Geometric Topology, Tome 20 (2020) no. 1, pp. 279-301. doi: 10.2140/agt.2020.20.279

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