Vol. 9, No. 5, 2016

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ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
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 SL2(). The set of nonabelian SL2() representations of a twist knot K is described as the zero set in × of a polynomial PK(x,y) = QK(y) + x2RK(y) [x,y], where x is the trace of a meridian. We prove some properties of PK(x,y). In particular, we prove that PK(2,y) [y] is irreducible over . 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