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

Abstract

We describe a procedure for computing the $\mathrm{SL}(2,\mathbb{C})$-character variety of an arborescent knot. Along the way, we clarify several facts about representations of arborescent tangles. Then we study a family of hyperbolic knots whose exteriors contain closed essential surfaces, showing that each of these knots has $1$-dimensional character variety. This provides infinitely many positive answers to a question of Boyer and Zhang posed in 1998.

24 pages, 7 figures. Compared with the previous version, several parts of content are replaced