Parabolic representations and generalized Riley polynomials

TL;DR

This paper extends Riley's work on parabolic representations from two-bridge knots to all knots in $S^3$, introducing generalized Riley polynomials and quandle methods to classify representations and compute complex volumes.

Contribution

It introduces generalized Riley polynomials and a new $u$-polynomial for all knots, providing explicit classification of parabolic representations and their invariants.

Findings

Explicit formulas for parabolic representations and complex volumes.

Complete classification of parabolic representations for knots up to 12 crossings.

Introduction of Riley and $u$-fields related to trace fields.

Abstract

We generalize R. Riley's study about parabolic representations of two bridge knot groups to the general knots in $S^3$. We utilize the parabolic quandle method for general knot diagrams and adopt symplectic quandle for better investigation, which gives such representations and their complex volumes explicitly. For any knot diagram with a specified crossing $c$, we define a generalized Riley polynomial $R_c(y) \in \mathbb{Q}[y]$ whose roots correspond to the conjugacy classes of parabolic representations of the knot group. The sign-type of parabolic quandle is newly introduced and we obtain a formula for the obstruction class to lift to a boundary unipotent $\text{SL}_2 \mathbb{C}$-representation. Moreover, we define another polynomial $g_c(u)\in\mathbb{Q}[u]$, called $u$-polynomial, and prove that $R_c(u^2)=\pm g_c(u)g_c(-u)$. Based on this result, we introduce and investigate Riley field and $u$-field which are closely related to the invariant trace field. This method eventually leads to the complete classification of parabolic representations of knot groups along with their complex volumes and cusp shapes up to 12 crossings.