Vol. 295, No. 2, 2018

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ISSN: 0030-8730
Quandle theory and the optimistic limits of the representations of link groups

Jinseok Cho

Vol. 295 (2018), No. 2, 329–366
Abstract

For a given boundary-parabolic representation of a link group to PSL(2, ), Inoue and Kabaya suggested a combinatorial method to obtain the developing map of the representation using the octahedral triangulation and the shadow-coloring of certain quandles. A quandle is an algebraic system closely related to the Reidemeister moves, so their method changes quite naturally under the Reidemeister moves.

We apply their method to the potential function, which was used to define the optimistic limit, and construct a saddle point of the function. This construction works for any boundary-parabolic representation, and it shows that the octahedral triangulation is good enough to study all possible boundary-parabolic representations of the link group. Furthermore, the evaluation of the potential function at the saddle point becomes the complex volume of the representation, and this saddle point changes naturally under the Reidemeister moves because it is constructed using the quandle.

Keywords
optimistic limit, quandle, hyperbolic volume, boundary-parabolic representation, link group
Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 51M25, 58J28
Milestones
Received: 16 January 2016
Revised: 30 January 2018
Accepted: 31 January 2018
Published: 11 April 2018
Authors
Jinseok Cho
Busan National University of Education
Busan
South Korea