Representation spaces of pretzel knots

TL;DR

This paper investigates the structure of representation spaces associated with certain pretzel knots, revealing their non-degenerate nature and the presence of non-binary dihedral SU(2) representations, contributing to knot theory and Floer homology.

Contribution

It provides a detailed analysis of the representation spaces for pretzel knots, especially those with pairwise coprime parameters, highlighting their non-degeneracy and novel SU(2) representations.

Findings

Representation spaces are non-degenerate for certain pretzel knots.

Existence of SU(2) representations that are not binary dihedral.

Insights into the structure of knot Floer homology for pretzel knots.

Abstract

We study the representation spaces $R(K;\bf{i})$ as appearing in Kronheimer and Mrowka's framed instanton knot Floer homology, for a class of pretzel knots. In particular, for pretzel knots $P(p,q,r)$ with $p, q, r$ pairwise coprime, these appear to be non-degenerate and comprise representations in SU(2) that are not binary dihedral.