Singular instanton homology of dual knots

Abstract

We establish a dimension formula for the unreduced singular instanton homology of dual knots ${\tilde{K}}_{p/q}\subset S_{p/q}^3(K)$ for a knot $K\subset S^3$: $$\dim I^{♯}(S_{p/q}^3(K),{\tilde{K}}_{p/q},\ddot{I}‰;\mathbb{K})=2q\cdot r_{\mathbb{K}}(K)+2|p-q\cdot \hat{I}{½}_{\mathbb{K}}^{♯}(K)|\mathrm{for}p/q\ne \hat{I}{½}_{\mathbb{K}}^{♯}(K),$$where $\ddot{I}‰\subset S^3\backslash K$ is any unoriented $1$-submanifold as the bundle set, $r_{\mathbb{K}}(K)$ and $\hat{I}{½}_{\mathbb{K}}^{♯}(K)$ are integers from the dimension formula of $I^{♯}(S_{p/q}^3(K);\mathbb{K})$ 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\[\dim I^\natural(S^3_{p/q}(K),\widetilde{K}_{p/q},ω;\mathbb{F}_2)=\dim I^\sharp(S^3_{p/q}(K);\mathbb{F}_2)~\mathrm{for}~p/q\neq ν^\sharp_{\mathbb{F}_2}(K).\]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 $p/q\in (0,6)$ with $p$ odd prime power, the surgery manifold ${\hat{Y}}_{p/2q}(\hat{K})$ is not $SU(2)$-abelian for the double branched cover $\hat{Y}=\hat{I}\pounds (S^3,K)$ and the preimage $\hat{K}\subset \hat{Y}$ 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.

15 pages, no figures; Comments are welcome