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Character varieties of even classical pretzel knots

Haimiao Chen

Abstract

For each even classical pretzel knot $P(2k_1+1,2k_2+1,2k_3)$, we determine the character variety of irreducible ${\rm SL}(2,\mathbb{C})$-representations, and clarify the steps of computing its A-polynomial.

Character varieties of even classical pretzel knots

Abstract

For each even classical pretzel knot , we determine the character variety of irreducible -representations, and clarify the steps of computing its A-polynomial.

Paper Structure

This paper contains 4 sections, 8 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Irreducible representations
  4. On the A-polynomial

Key Result

Theorem 1.1

The irreducible character variety of the even pretzel knot $P(2k_1+1,2k_1+1,2k_3)$ can be embedded in and is the disjoint union of four parts: $\mathcal{X}^{\rm irr}(K)=\mathcal{X}_0\sqcup\mathcal{X}_1\sqcup\mathcal{X}_2\sqcup\mathcal{X}_3,$ where The dimensions are: $\dim\mathcal{X}_{0}=\dim\mathcal{X}_{1}=0$, $\dim\mathcal{X}_2=\dim\mathcal{X}_3=1$.

Figures (1)

  • Figure 1: The pretzel knot $P(2k_1+1,2k_2+1,2k_3)$, with $k_1=1, k_2=k_3=2$

Theorems & Definitions (12)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • proof
  • Lemma 3.5
  • ...and 2 more