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Character varieties of odd classical pretzel knots

Haimiao Chen

TL;DR

This paper determines the ${\rm SL}(2,\mathbb{C})$-character variety ${\mathcal X}^{\rm irr}(K)$ for odd classical pretzel knots $K=P(2k_1+1,2k_2+1,2k_3+1)$ by translating matrix relations into trace identities and classifying reducible vs irreducible representations. The irreducible locus is shown to decompose into four strata, including a finite set of points, a family of conics, and a high-genus algebraic curve, with a comprehensive parametrization in terms of trace data and auxiliary variables. From this structure, the paper outlines a method to compute the $A$-polynomial, extracting a “hard” factor from the $\mathcal X_3$ component while accounting for the conic factors coming from $\mathcal X_2$. The approach provides a concrete elimination-based framework for obtaining $A_K$ and demonstrates its feasibility in a concrete example, enhancing understanding of how character varieties encode knot and 3-manifold topology and contributing to AJ-type conjectures and exceptional filling analysis.

Abstract

We determine the ${\rm SL}(2,\mathbb{C})$-character variety for each odd classical pretzel knot $P(2k_1+1,2k_2+1,2k_3+1)$, and present a method for computing its A-polynomial.

Character varieties of odd classical pretzel knots

TL;DR

This paper determines the -character variety for odd classical pretzel knots by translating matrix relations into trace identities and classifying reducible vs irreducible representations. The irreducible locus is shown to decompose into four strata, including a finite set of points, a family of conics, and a high-genus algebraic curve, with a comprehensive parametrization in terms of trace data and auxiliary variables. From this structure, the paper outlines a method to compute the -polynomial, extracting a “hard” factor from the component while accounting for the conic factors coming from . The approach provides a concrete elimination-based framework for obtaining and demonstrates its feasibility in a concrete example, enhancing understanding of how character varieties encode knot and 3-manifold topology and contributing to AJ-type conjectures and exceptional filling analysis.

Abstract

We determine the -character variety for each odd classical pretzel knot , and present a method for computing its A-polynomial.

Paper Structure

This paper contains 7 sections, 9 theorems, 71 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The Character variety
  4. Set up
  5. Irreducible representations
  6. The character variety
  7. The A-polynomial

Key Result

Lemma 2.1

For any $X,Y\in{\rm SL}(2,\mathbb{C})$ with ${\rm tr}(X)=t_{1}$, ${\rm tr}(Y)=t_{2}$ and ${\rm tr}(XY)=t_{12}$, one has

Figures (2)

  • Figure 1: The tangle as a part of a link
  • Figure 2: The pretzel knot $P(3,5,5)$

Theorems & Definitions (23)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Remark 3.1
  • Remark 3.3
  • Remark 3.4
  • Lemma 3.5
  • proof
  • Remark 3.6
  • Proposition 3.7
  • ...and 13 more