Singular instanton homology of dual knots
Fan Ye
TL;DR
This work determines a dimension formula for the unreduced singular instanton homology of dual knots under Dehn surgery: for a knot $K\subset S^3$ and coprime $p/q$, with $M=\nu^{\sharp}_{\mathbb{K}}(K)$ and $R=r_{\mathbb{K}}(K)$, the dual knot invariant satisfies $\dim I^\sharp(S^3_{p/q}(K),{\widetilde{K}}_{p/q},\omega;\mathbb{K}) = 2qR+2|p-qM|$ when $p/q\neq M$, with an exceptional case at $p/q=M$; over $\mathbb{F}_2$ the reduced dimension matches the unreduced one for $p/q\neq M$. The authors develop a comprehensive framework for instanton knot invariants, including four interrelated definitions of instanton homology, their dependence on the bundle set $\omega$, and cobordism naturality, then prove the dual-knot dimension formula via integral-to-rational slope analysis anchored by Bhat’s triangle. They further connect these dimension counts to $SU(2)$-representation theory, showing non-abelian representations for a broad class of slopes and knots, including determinant-one knots and double branched covers, thereby enriching the understanding of instanton L-space knots and providing nontrivial obstructions to $SU(2)$-abelian structures. Overall, the results offer a robust tool for certifying non-abelian $SU(2)$ representations and for analyzing the behavior of instanton homology under dual knots and Dehn surgeries.
Abstract
We establish a dimension formula for the unreduced singular instanton homology of dual knots $\widetilde{K}_{p/q}\subset S^3_{p/q}(K)$ for a knot $K\subset S^3$: $$ \dim I^\sharp(S^3_{p/q}(K),\widetilde{K}_{p/q},ω; \mathbb{K}) = 2q \cdot r_{\mathbb{K}}(K) + 2|p - q \cdot ν^\sharp_{\mathbb{K}}(K)|~\mathrm{for}~p/q\neq ν^\sharp_{\mathbb{K}}(K), $$where $ω\subset S^3\backslash K$ is any unoriented $1$-submanifold as the bundle set, $r_{\mathbb{K}}(K)$ and $ν^\sharp_{\mathbb{K}}(K)$ are integers from the dimension formula of $I^\sharp(S^3_{p/q}(K);\mathbb{K})$ 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 $\widehat{Y}_{p/2q}(\widehat{K})$ is not $SU(2)$-abelian for the double branched cover $\widehat{Y}=Σ(S^3,K)$ 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.