On nonabelian representations of twist knots

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 × C of a polynomial P K (x, y) = Q K (y) + x 2 R K (y) ∈ 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) ∈ 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.