An enhanced Euler characteristic of sutured instanton homology
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2021
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| _version_ | 1866917674407165952 |
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| author | Li, Zhenkun Ye, Fan |
| author_facet | Li, Zhenkun Ye, Fan |
| contents | For a balanced sutured manifold $(M,γ)$, we construct a decomposition of $SHI(M,γ)$ with respect to torsions in $H=H_1(M;\mathbb{Z})$, which generalizes the decomposition of $I^\sharp(Y)$ in previous work of the authors. This decomposition can be regarded as a candidate for the counterpart of the torsion spin$^c$ decompositions in $SFH(M,γ)$. Based on this decomposition, we define an enhanced Euler characteristic $χ_{\rm en}(SHI(M,γ))\in\mathbb{Z}[H]/\pm H$ and prove that $χ_{\rm en}(SHI(M,γ))=χ(SFH(M,γ))$. This provides a better lower bound on $\dim_\mathbb{C}SHI(M,γ)$ than the graded Euler characteristic $χ_{\rm gr}(SHI(M,γ))$. As applications, we prove instanton knot homology detects the unknot in any instanton L-space and show that the conjecture $KHI(Y,K)\cong \widehat{HFK}(Y,K)$ holds for all $(1,1)$-L-space knots and constrained knots in lens spaces, which include all torus knots and many hyperbolic knots in lens spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_10490 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | An enhanced Euler characteristic of sutured instanton homology Li, Zhenkun Ye, Fan Geometric Topology For a balanced sutured manifold $(M,γ)$, we construct a decomposition of $SHI(M,γ)$ with respect to torsions in $H=H_1(M;\mathbb{Z})$, which generalizes the decomposition of $I^\sharp(Y)$ in previous work of the authors. This decomposition can be regarded as a candidate for the counterpart of the torsion spin$^c$ decompositions in $SFH(M,γ)$. Based on this decomposition, we define an enhanced Euler characteristic $χ_{\rm en}(SHI(M,γ))\in\mathbb{Z}[H]/\pm H$ and prove that $χ_{\rm en}(SHI(M,γ))=χ(SFH(M,γ))$. This provides a better lower bound on $\dim_\mathbb{C}SHI(M,γ)$ than the graded Euler characteristic $χ_{\rm gr}(SHI(M,γ))$. As applications, we prove instanton knot homology detects the unknot in any instanton L-space and show that the conjecture $KHI(Y,K)\cong \widehat{HFK}(Y,K)$ holds for all $(1,1)$-L-space knots and constrained knots in lens spaces, which include all torus knots and many hyperbolic knots in lens spaces. |
| title | An enhanced Euler characteristic of sutured instanton homology |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2107.10490 |