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

TL;DR

This paper establishes a criterion linking the Alexander polynomial's roots to the existence of continuous families of irreducible SL(2,R) representations of knot groups, with applications to L-space knots.

Contribution

It proves a new criterion connecting Alexander polynomial roots to SL(2,R) representations and applies it to show L-space knot groups admit such representations.

Findings

Knots with Alexander polynomial roots of odd order on the unit circle have associated irreducible SL(2,R) representations.

Nontrivial L-space knots satisfy the criterion, implying their groups admit irreducible SL(2,R) representations.

The criterion provides a new link between knot invariants and representation theory.

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.