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× π1(R) to π1(G), where R is a subvariety of the variety of representations A → G. In the case where A is the peripheral subgroup of a torus or twobridge knot group, G = S1 and 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.