Trace-free ${\rm SL}(2,\mathbb{C})$-representations of arborescent links

TL;DR

This paper develops a method to classify trace-free ${ m SL}(2,bC)$-representations of arborescent links and applies it to specific 3-bridge cases, advancing understanding of link group representations.

Contribution

It introduces a comprehensive technique for determining trace-free ${ m SL}(2,bC)$-representations of arborescent links, including explicit computations for certain 3-bridge links.

Findings

Complete classification method for trace-free representations

Explicit computations for a class of 3-bridge arborescent links

Enhanced understanding of link group representations

Abstract

Given a link $L\subset S^3$, a representation $\pi_1(S^3-L)\to{\rm SL}(2,\mathbb{C})$ is {\it trace-free} if it sends each meridian to an element with trace zero. We present a method for completely determining trace-free ${\rm SL}(2,\mathbb{C})$-representations for arborescent links. Concrete computations are done for a class of 3-bridge arborescent links.