#### Vol. 9, No. 5, 2016

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On nonabelian representations of twist knots

### James C. Dean and Anh T. Tran

Vol. 9 (2016), No. 5, 831–838
##### 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 $ℂ×ℂ$ of a polynomial ${P}_{K}\left(x,y\right)={Q}_{K}\left(y\right)+{x}^{2}{R}_{K}\left(y\right)\in ℤ\left[x,y\right]$, where $x$ is the trace of a meridian. We prove some properties of ${P}_{K}\left(x,y\right)$. In particular, we prove that ${P}_{K}\left(2,y\right)\in ℤ\left[y\right]$ is irreducible over $ℚ$. 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.

##### Keywords
Chebychev polynomial, nonabelian representation, parabolic representation, trace field, twist knot
Primary: 57N10
Secondary: 57M25