Symplectic quandle Method and $SL(2,\mathbb C)$-representations of 2-bridge Knots

TL;DR

This paper extends the symplectic quandle method to analyze all $SL(2,b C)$-representations of 2-bridge knots, simplifying calculations of Riley and A-polynomials and enabling rapid computation of many new invariants.

Contribution

It introduces a generalized symplectic quandle structure for $SL(2,b C)$-representations of 2-bridge knot groups, simplifying the computation of key polynomial invariants.

Findings

Derived a simpler expression for the Riley polynomial.

Developed recursive formulas for Riley and Alexander polynomials.

Enabled rapid computation of numerous new A-polynomials.

Abstract

In this paper, we extend the symplectic quandle method, previously employed in our study of parabolic representations of knot groups, to investigate the general $SL(2,\mathbb{C})$-representations of 2-bridge ``kmot" groups. We introduce a `generalized symplectic quandle structure' corresponding to ($\mathcal{D}_M$, conjugation) for each $M\in\mathbb C\setminus \{0,1,-1\}$, where $\mathcal{D}_M=\{A\in SL(2,\mathbb{C})\mid tr(A)= M+M^{-1} \}$. By converting the system of conjugation quandle equations to that of generalized symplectic quandle equations, we obtain a simpler expression for the 2-variable Riley polynomial and derive some recursive formulas for Riley polynomials and Alexander polynomials. This approach enables us to effectively compute the A-polynomials, allowing us to obtain numerous previously unknown A-polynomials within minutes using Mathematica.