On nonabelian representations of twist knots

Dean, James C.; Tran, Anh T.

doi:10.2140/involve.2016.9.831

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On nonabelian representations of twist knots

James C. Dean and Anh T. Tran

Vol. 9 (2016), No. 5, 831–838

DOI: 10.2140/involve.2016.9.831

Abstract

We study representations of the knot groups of twist knots into ${SL}_{2} (ℂ)$ . The set of nonabelian ${SL}_{2} (ℂ)$ representations of a twist knot $K$ is described as the zero set in $ℂ \times ℂ$ of a polynomial $P_{K} (x, y) = Q_{K} (y) + x^{2} R_{K} (y) \in ℤ [x, y]$ , where $x$ is the trace of a meridian. We prove some properties of $P_{K} (x, y)$ . In particular, we prove that $P_{K} (2, y) \in ℤ [y]$ 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

Mathematical Subject Classification 2010

Primary: 57N10

Secondary: 57M25

Milestones

Received: 6 July 2015

Revised: 30 October 2015

Accepted: 3 November 2015

Published: 25 August 2016

Communicated by Jim Hoste

Authors

James C. Dean
University of Dallas Irving, TX 75062 United States

Anh T. Tran
Department of Mathematical Sciences The University of Texas at Dallas Richardson, TX 75080 United States

Vol. 9, No. 5, 2016