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

Kyeonghee Jo, Hyuk Kim

TL;DR

This work generalizes the symplectic quandle method to non-abelian $SL(2,\mathbb{C})$-representations of 2-bridge kmot groups by introducing a family of generalized symplectic quandle structures parameterized by $M\in\mathbb{C}\setminus\{0,1,-1\}$. By translating conjugation quandle equations into generalized symplectic quandle equations, it yields a streamlined two-variable Riley polynomial and recursive relations for Riley and Alexander polynomials, enabling efficient computation of A-polynomials. The approach reveals a close link to Chebyshev polynomials and provides practical tools to compute A-polynomials for many 2-bridge knots (up to 12 crossings) with previously unknown polynomials, using Mathematica. Overall, the paper unifies parabolic and general representations within a single framework and enhances the computational study of knot group character varieties and 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.

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

TL;DR

This work generalizes the symplectic quandle method to non-abelian -representations of 2-bridge kmot groups by introducing a family of generalized symplectic quandle structures parameterized by . By translating conjugation quandle equations into generalized symplectic quandle equations, it yields a streamlined two-variable Riley polynomial and recursive relations for Riley and Alexander polynomials, enabling efficient computation of A-polynomials. The approach reveals a close link to Chebyshev polynomials and provides practical tools to compute A-polynomials for many 2-bridge knots (up to 12 crossings) with previously unknown polynomials, using Mathematica. Overall, the paper unifies parabolic and general representations within a single framework and enhances the computational study of knot group character varieties and 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 -representations of 2-bridge ``kmot" groups. We introduce a `generalized symplectic quandle structure' corresponding to (, conjugation) for each , where . 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.
Paper Structure (7 sections, 34 theorems, 187 equations)

This paper contains 7 sections, 34 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Conjugation quandle structure on $SL(2,{\mathbb C})$
  3. $\epsilon_i$-sequences and $c_k^n, d_k^n, \tilde{c}_k^n, \tilde{d}_k^n$
  4. Non-abelian $SL(2,{\mathbb C})$-representations of kmot groups
  5. Symplectic quandle method
  6. Recursive formulas for Riley polynomials and Alexander polynomials
  7. 2-bridge knots

Key Result

Proposition 2.1

If $T(A)=[a]$ and $T(B)=[b]$ then

Theorems & Definitions (70)

  • Proposition 2.1: Jo-Kim
  • Lemma 2.2: Riley3
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Corollary 2.7
  • Lemma 2.8
  • ...and 60 more