On nonabelian representations of twist knots

Abstract

We study representations of the knot groups of twist knots into ${SL}_2\left(ℂ\right)$. The set of nonabelian ${SL}_2\left(ℂ\right)$ representations of a twist knot $K$ is described as the zero set in $ℂ\times ℂ$ of a polynomial $P_K\left(x,y\right)=Q_K\left(y\right)+x^2{R}_K\left(y\right)\in \mathbb{Z}\left[x,y\right]$, where $x$ is the trace of a meridian. We prove some properties of $P_K\left(x,y\right)$. In particular, we prove that $P_K\left(2,y\right)\in \mathbb{Z}\left[y\right]$ is irreducible over $\mathbb{Q}$. As a consequence, we obtain an alternative proof of a result of Hoste and Shanahan that the degree of the trace field is precisely two less than the minimal crossing number of a twist knot.