Suppose that $K$ is an $\textit{alternating}$ knot in $S^3$, and let $R_0$ be the space of homomorphisms from $\pi_1(S^3 - K)\to SU(2)$ which send meridians to trace free matrices. Denote the subset of $R_0$ consisting of metabelian representations as $R_m \subset R_0$.

Question: when $K$ is prime, is there any reason to think that $R_0$ retracts to $R_m$? Could at least $H_*(R_0;\mathbb{Z})=H_*(R_m;\mathbb{Z})$? Does anyone know of anything about such questions in the literature?

Might there be similar analyses for more general $SU(2)$ representations of knot groups?

deformationretract (this is suggested by your $H_*$ question): generically the space of trace free reps (mod conjugation) is finite and so there would obviously be a retract. – Paul Mar 18 '10 at 3:13