Singular instanton homology of dual knots

We establish a dimension formula for the unreduced singular instanton homology of dual knots ${\tilde{K}}_{p\mathrm{/}q}\subset S_{p\mathrm{/}q}^3(K)$ for a knot $K\subset S^3$: $$\dim I^{\mathrm{♯}}(S_{p\mathrm{/}q}^3(K),{\tilde{K}}_{p\mathrm{/}q},\omega ;\mathbb{K})=2q\cdot r_{\mathbb{K}}(K)+2\mathrm{\mid }p-q\cdot {\nu }_{\mathbb{K}}^{\mathrm{♯}}(K)\mathrm{\mid }\mathrm{f}\mathrm{o}\mathrm{r}p\mathrm{/}q\mathrm{\ne }{\nu }_{\mathbb{K}}^{\mathrm{♯}}(K),$$where $\omega \subset S^3\mathrm{\backslash }K$ is any unoriented $1$-submanifold as the bundle set, $r_{\mathbb{K}}(K)$ and ${\nu }_{\mathbb{K}}^{\mathrm{♯}}(K)$ are integers from the dimension formula of $I^{\mathrm{♯}}(S_{p\mathrm{/}q}^3(K);\mathbb{K})$ for a field $\mathbb{K}$ defined by Li and the author. In particular, when $\mathbb{K}$ is the two-element field ${\mathbb{F}}_2$, the reduced singular instanton homology satisfies$$\dim I^{\mathrm{♮}}(S_{p\mathrm{/}q}^3(K),{\tilde{K}}_{p\mathrm{/}q},\omega ;{\mathbb{F}}_2)=\dim I^{\mathrm{♯}}(S_{p\mathrm{/}q}^3(K);{\mathbb{F}}_2)\mathrm{f}\mathrm{o}\mathrm{r}p\mathrm{/}q\mathrm{\ne }{\nu }_{{\mathbb{F}}_2}^{\mathrm{♯}}(K)\mathrm{.}$$As an application, for a determinant-one knot $K\subset S^3$ other than the unknot and the torus knots $T_{2,3},T_{2,5}$ and a rational $p\mathrm{/}q\in (0,6)$ with $p$ odd prime power, the surgery manifold ${\widehat{Y}}_{p\mathrm{/}2q}(\widehat{K})$ is not $SU(2)$-abelian for the double branched cover $\widehat{Y}=\mathrm{\Sigma }(S^3,K)$ and the preimage $\widehat{K}\subset \widehat{Y}$ of $K$. We also obtain non-abelian results for $SU(2)$ representations of the knot complement that send the curves of some fixed slope in $(0,6)$ to traceless elements.