Winding numbers and SU(2)-representations of knot groups

TL;DR

This paper introduces a bilinear pairing connecting abelian groups, knot group representations, and SU(2)-representations, revealing non-trivial interactions with implications for knot theory and 3-manifold topology.

Contribution

It constructs a new bilinear pairing involving knot group representations and demonstrates its non-triviality for specific knot classes, advancing understanding of SU(2)-representations.

Findings

The pairing is non-zero for certain knot groups.

Implications for SU(2)-representations of manifolds from Dehn surgery.

New tools for studying knot group representations.

Abstract

Given an abelian group $A$ and a Lie group $G$, we construct a bilinear pairing from $A\times\pi_1({\mathcal R})$ to $\pi_1(G)$, where $\mathcal R$ is a subvariety of the variety of representations $A\to G$. In the case where $A$ is the peripheral subgroup of a torus or two-bridge knot group, $G=S^1$ and $\mathcal R$ is a certain variety of representations arising from suitable SU(2)-representations of the knot group, we show that this pairing is not identically zero. We discuss the consequences of this result for the SU(2)-representations of fundamental groups of manifolds obtained by Dehn surgery on such knots.