Deforming abelian elliptic $\mathrm{SL}(2,\mathbb{R})$--representations of knot groups

Abstract

The following criterion is proved in this paper. If the Alexander polynomial of a knot $K\subset S^3$ has a zero of odd order on the complex unit circle, then there exists a continuous family of irreducible representations $\pi_1(S^3\setminus K)\to \mathrm{SL}(2,\mathbb{R})$ converging to an abelian representation of noncentral elliptic type. As an application, the author shows that the Alexander polynomial of any nontrivial L-space knot satisfies the condition of the criterion. In particular, it follows that the fundamental group of any nontrivial L-space knot complement admits an irreducible $\mathrm{SL}(2,\mathbb{R})$--representation.