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?



SU(2) representations of alternating knot groups
I doubt there is such a retraction. Such representations are representations of the $\pi$-orbifold, obtained by killing the square of the meridian (at least if one quotients by $\pm Id$). If one takes a Montesinos knot, these orbifolds are Seifert fibered, and such representations should factor through the quotient orbifold. Such oribfolds can have several non-metabelian isolated representations into $SO(3)$, so I expect that the answer is no even on $H\_0$. 


