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Tracefree ${\rm SL}(2,\mathbb{C})$-representations of Montesinos links

Haimiao Chen

Abstract

Given a link $L$, a representation $π_1(S^{3}-L)\to {\rm SL}(2,\mathbb{C})$ is {\it tracefree} if the image of each meridian has trace zero. We determine the conjugacy classes of tracefree representations when $L$ is a Montesinos link.

Tracefree ${\rm SL}(2,\mathbb{C})$-representations of Montesinos links

Abstract

Given a link , a representation is {\it tracefree} if the image of each meridian has trace zero. We determine the conjugacy classes of tracefree representations when is a Montesinos link.

Paper Structure

This paper contains 3 sections, 2 theorems, 44 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Representations of rational tangles
  3. Representations of Montesinos links

Key Result

Lemma 2.1

(i) If $\rho$ is a representation of $[k]$ with $\rho^{{\rm nw}}=A(a_1)$ and $\rho^{{\rm sw}}=A(a_2)$, then $\rho^{{\rm ne}}=A(-a_1^{k+1}/a_2^k)$ and $\rho^{{\rm se}}=A(-a_1^{k}/a_2^{k-1})$. (ii) If $\rho$ is a representation of $[1/k]$ with $\rho^{{\rm nw}}=A(b_1)$ and $\rho^{{\rm ne}}=A(b_2)$, the

Figures (5)

  • Figure 1: A representation satisfies $\rho(z)=\rho(x)\rho(y)\rho(x)^{-1}$ at each crossing
  • Figure 2: A tangle $T\in\mathcal{T}_2^2$, with the four ends directed outwards
  • Figure 3: The simplest four tangles: (a) $[0]$, (b) $[\infty]$, (c) $[1]$, (d) $[-1]$
  • Figure 4: (a) $T_1\ast T_2$; (b) $T_1\star T_2$
  • Figure 5: A representation of the rational tangle $[[k_{1}],[k_{2}],[k_{3}]]$

Theorems & Definitions (8)

  • Remark 1.1
  • Lemma 2.1
  • proof
  • Remark 2.3
  • Remark 3.1
  • Theorem 3.2
  • Remark 3.3
  • Remark 3.4