Deforming representations of knot groups in \(SU(2)\) (Q1179509)

In an earlier paper the second author showed that if an abelian representation of a knot group in \(SU(2)\) is the limit of nonabelian representations then the image of the square of a meridian is a root of the Alexander polynomial of the knot [Trans. Am. Math. Soc. 326, No. 2, 795-828 (1991)]. This paper provides a partial converse by showing that abelian representations corresponding to simple roots of the Alexander polynomial are endpoints of arcs of nonabelian representations. A similar result holds also for representations in \(SL(2,\mathbb{R})\). The key ideas are that these matrix groups double cover the isometry groups of surfaces of constant curvature \(c>0\) and \(c<0\), respectively, and that roots of the Alexander polynomial determine nonabelian representations of the knot group in the isometry group of the plane (\(c=0\)), an observation originally due to De Rham.