  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.
