Khovanov homology and exotic $4$-manifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918253018742784 |
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| author | Ren, Qiuyu Willis, Michael |
| author_facet | Ren, Qiuyu Willis, Michael |
| contents | We show that the Khovanov-Rozansky $\mathfrak{gl}_2$ skein lasagna module distinguishes the exotic pair of knot traces $X_{-1}(-5_2)$ and $X_{-1}(P(3,-3,-8))$, an example first discovered by Akbulut. This gives the first analysis-free proof of the existence of exotic compact orientable $4$-manifolds. We also present a family of exotic knot traces that seem not directly recoverable from gauge/Floer-theoretic methods. Along the way, we present new explicit calculations of the Khovanov skein lasagna modules, and we define lasagna generalizations of the Lee homology and Rasmussen $s$-invariant, which are of independent interest. Other consequences of our work include a slice obstruction of knots in $4$-manifolds with nonvanishing skein lasagna module, a sharp shake genus bound for some knots from the lasagna $s$-invariant, and a construction of induced maps on Khovanov homology for cobordisms in $k\mathbb{CP}^2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_10452 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Khovanov homology and exotic $4$-manifolds Ren, Qiuyu Willis, Michael Geometric Topology Quantum Algebra 57K41 (Primary) 57K18, 57R55, 57R56 (Secondary) We show that the Khovanov-Rozansky $\mathfrak{gl}_2$ skein lasagna module distinguishes the exotic pair of knot traces $X_{-1}(-5_2)$ and $X_{-1}(P(3,-3,-8))$, an example first discovered by Akbulut. This gives the first analysis-free proof of the existence of exotic compact orientable $4$-manifolds. We also present a family of exotic knot traces that seem not directly recoverable from gauge/Floer-theoretic methods. Along the way, we present new explicit calculations of the Khovanov skein lasagna modules, and we define lasagna generalizations of the Lee homology and Rasmussen $s$-invariant, which are of independent interest. Other consequences of our work include a slice obstruction of knots in $4$-manifolds with nonvanishing skein lasagna module, a sharp shake genus bound for some knots from the lasagna $s$-invariant, and a construction of induced maps on Khovanov homology for cobordisms in $k\mathbb{CP}^2$. |
| title | Khovanov homology and exotic $4$-manifolds |
| topic | Geometric Topology Quantum Algebra 57K41 (Primary) 57K18, 57R55, 57R56 (Secondary) |
| url | https://arxiv.org/abs/2402.10452 |