Singular instanton homology of dual knots

TL;DR

This paper derives a formula for the dimension of singular instanton homology of dual knots in Dehn surgeries, revealing new invariants and non-abelian properties of certain knot complements.

Contribution

It establishes a dimension formula for singular instanton homology of dual knots, connecting it to known invariants and applying it to non-abelian $SU(2)$ representations.

Findings

Dimension formula for singular instanton homology of dual knots.

Equality of reduced and unreduced homology dimensions under certain conditions.

Non-abelian $SU(2)$ representation results for specific knot 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 $K\subset S^3$: $$ \dim I^\sharp(S^3_{p/q}(K),\widetilde{K}_{p/q},\omega; \mathbb{K}) = 2q \cdot r_{\mathbb{K}}(K) + 2|p - q \cdot \nu^\sharp_{\mathbb{K}}(K)|~\mathrm{for}~p/q\neq \nu^\sharp_{\mathbb{K}}(K), $$where $\omega\subset S^3\backslash K$ is any unoriented $1$-submanifold as the bundle set, $r_{\mathbb{K}}(K)$ and $\nu^\sharp_{\mathbb{K}}(K)$ 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\[\dim I^\natural(S^3_{p/q}(K),\widetilde{K}_{p/q},\omega;\mathbb{F}_2)=\dim I^\sharp(S^3_{p/q}(K);\mathbb{F}_2)~\mathrm{for}~p/q\neq \nu^\sharp_{\mathbb{F}_2}(K).\]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 $\widehat{Y}=\Sigma(S^3,K)$ 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.