Singular instanton homology of dual knots
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918217520250880 |
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| author | Ye, Fan |
| author_facet | Ye, Fan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19883 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Singular instanton homology of dual knots Ye, Fan Geometric Topology 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. |
| title | Singular instanton homology of dual knots |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2511.19883 |