In this article, using 3-orbifolds singular along a knot with underlying space a homology sphere $Y^3$, the question of existence of non-trivial and non-abelian $\SU(2)$-representations of the fundamental group of cyclic branched covers of $Y^3$ along a knot is studied. We first use Floer Homology for knots to derive an existence result of non-abelian $\SU(2)$-representations of the fundamental group of knot complements, for knots with a non-vanishing equivariant signature. This provides information on the existence of non-trivial and non-abelian $\SU(2)$-representations of the fundamental group of cyclic branched covers. We illustrate the method with some examples of knots in $S^3$.

Let $\tilde M$ be a regular branched cover of a homology 3-sphere $M$ with deck group $G\cong \zt^d$ and branch set a trivalent graph $\Gamma$; such a cover is determined by a coloring of the edges of $\Gamma$ with elements of $G$. For each index-2 subgroup $H$ of $G$, $M_H = \tilde M/H$ is a double branched cover of $M$. Sakuma has proved that $H_1(\tilde M)$ is isomorphic, modulo 2-torsion, to $\bigoplus_H H_1(M_H)$, and has shown that $H_1(\tilde M)$ is determined up to isomorphism by $\bigoplus_H H_1(M_H)$ in certain cases; specifically, when $d=2$ and the coloring is such that the branch set of each cover $M_H\to M$ is connected, and when $d=3$ and $\Gamma$ is the complete graph $K_4$. We prove this for a larger class of coverings: when $d=2$, for any coloring of a connected graph; when $d=3$ or $4$, for an infinite class of colored graphs; and when $d=5$, for a single coloring of the Petersen graph.