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

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.