The ghost character of the (4,5)-torus knot and its applications

TL;DR

This paper demonstrates that the (4,5)-torus knot has a unique ghost character, which reveals new insights into the representations of its 2-fold branched cover's fundamental group and provides a counterexample to Ng's conjecture about character varieties.

Contribution

It introduces the concept of a ghost character for the (4,5)-torus knot and shows its implications for representation theory and character variety mappings, including a counterexample to Ng's conjecture.

Findings

The (4,5)-torus knot admits exactly one ghost character.

The ghost character explains the existence of certain $ ext{SL}_2( ext{C})$-representations not arising from trace-free representations.

The (4,5)-torus knot provides a counterexample to Ng's conjecture, with the map $h^*$ being surjective but not injective.

Abstract

We show that the (4,5)-torus knot $T_{4,5}$ admits exactly one ghost character. We then show that this ghost character provides the following two important results. (1) It is known that for any knot $K$ every (meridionally) trace-free $\SL_2(\C)$-representation of the knot group $G(K)$ yields an $\SL_2(\C)$-representation of the fundamental group $\pi_1(\Sigma_2K)$ of the 2-fold branched cover $\Sigma_2K$ of the 3-sphere along $K$. This correspondence often but not always provides all $\SL_2(\C)$-representations of $\pi_1(\Sigma_2K)$. We show by using the ghost character that $T_{4,5}$ is the simplest torus knot such that $\pi_1(\Sigma_2T_{4,5})$ admits an $\SL_2(\C)$-representation which cannot be realized by any trace-free $\SL_2(\C)$-representations. (2) We show that $T_{4,5}$ is the simplest torus knot that provides a counterexample to Ng's conjecture, concerned with a polynomial map $h^*$ between the character variety $X(\Sigma_2K)$ of $\pi_1(\Sigma_2K)$ and the fundamental variety $F_2(K)$. More precisely, the map $h^*$ is surjective but not injective, and hence not an isomorphism for $T_{4,5}$.