The ${\rm SL}(2,\mathbb{C})$-character variety of an arborescent knot

Abstract

We clarify steps for determining the ${\rm SL}(2,\mathbb{C})$-character variety of any arborescent knot. Interestingly, we show that the `excellent parts' of arborescent knots $K_1,K_2$ are isomorphic if $K_1$ can be related to $K_2$ through certain moves on projection diagrams. Furthermore, we give a sufficient condition in terms of diagram for the existence of components of dimension larger than $1$. A generalization to arborescent links is sketched.