Quandle theory and the optimistic limits of the representations of link groups

TL;DR

This paper extends quandle-based combinatorial methods to analyze boundary-parabolic representations of link groups, demonstrating that the octahedral triangulation captures all such representations and relates potential function evaluations to complex volumes.

Contribution

It applies quandle theory to construct saddle points of the potential function for any boundary-parabolic representation, linking geometric invariants to combinatorial structures.

Findings

Constructed saddle points for the potential function for all boundary-parabolic representations.

Showed that the octahedral triangulation suffices to study all boundary-parabolic representations.

Linked the potential function evaluation at saddle points to the complex volume of the representation.

Abstract

When a boudnary-parabolic representation of a link group to PSL(2,$\mathbb{C}$) is given, 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 quandle. Quandle is an algebraic system closely related with the Reidemeister moves, so their method changes quite naturally under the Reidemeister moves. In this article, we apply their method to the potential function, which was used to define the optimsitic 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.