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

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 ${\text{SL}}_2(\text{C})$-representation of the knot group $G(K)$ yields an ${\text{SL}}_2(\text{C})$-representation of the fundamental group ${\pi }_1({\mathrm{\Sigma }}_{2}K)$ of the 2-fold branched cover ${\mathrm{\Sigma }}_{2}K$ of the 3-sphere along $K$. This correspondence often but not always provides all ${\text{SL}}_2(\text{C})$-representations of ${\pi }_1({\mathrm{\Sigma }}_{2}K)$. We show by using the ghost character that $T_{4,5}$ is the simplest torus knot such that ${\pi }_1({\mathrm{\Sigma }}_2{T}_{4,5})$ admits an ${\text{SL}}_2(\text{C})$-representation which cannot be realized by any trace-free ${\text{SL}}_2(\text{C})$-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 $h^{\ast }$ between the character variety $X({\mathrm{\Sigma }}_{2}K)$ of ${\pi }_1({\mathrm{\Sigma }}_{2}K)$ and the fundamental variety $F_2(K)$. More precisely, the map $h^{\ast }$ is surjective but not injective, and hence not an isomorphism for $T_{4,5}$.