Kirby belts, categorified projectors, and the skein lasagna module of $S^{2}\times{S^{2}}$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912112495820800 |
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| author | Sullivan, Ian A. Zhang, Melissa |
| author_facet | Sullivan, Ian A. Zhang, Melissa |
| contents | We interpret Manolescu-Neithalath's cabled Khovanov homology formula for computing Morrison-Walker-Wedrich's $\mathrm{KhR}_2$ skein lasagna module as a homotopy colimit (mapping telescope) in a completion of the category of complexes over Bar-Natan's cobordism category. Using categorified projectors, we compute the $\mathrm{KhR}_2$ skein lasagna modules of (manifold, boundary link) pairs $(S^2 \times B^2, \tilde β)$, where $\tilde β$ is a geometrically essential boundary link, identifying a relationship between the lasagna module and the Rozansky projector appearing in the Rozansky-Willis invariant for nullhomologous links in $S^2 \times S^1$. As an application, we show that the $\mathrm{KhR}_2$ skein lasagna module of $S^2 \times S^2$ is trivial, confirming a conjecture of Manolescu. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_01081 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Kirby belts, categorified projectors, and the skein lasagna module of $S^{2}\times{S^{2}}$ Sullivan, Ian A. Zhang, Melissa Geometric Topology Quantum Algebra 57K18 (Primary) 57R56, 57K41 (Secondary) We interpret Manolescu-Neithalath's cabled Khovanov homology formula for computing Morrison-Walker-Wedrich's $\mathrm{KhR}_2$ skein lasagna module as a homotopy colimit (mapping telescope) in a completion of the category of complexes over Bar-Natan's cobordism category. Using categorified projectors, we compute the $\mathrm{KhR}_2$ skein lasagna modules of (manifold, boundary link) pairs $(S^2 \times B^2, \tilde β)$, where $\tilde β$ is a geometrically essential boundary link, identifying a relationship between the lasagna module and the Rozansky projector appearing in the Rozansky-Willis invariant for nullhomologous links in $S^2 \times S^1$. As an application, we show that the $\mathrm{KhR}_2$ skein lasagna module of $S^2 \times S^2$ is trivial, confirming a conjecture of Manolescu. |
| title | Kirby belts, categorified projectors, and the skein lasagna module of $S^{2}\times{S^{2}}$ |
| topic | Geometric Topology Quantum Algebra 57K18 (Primary) 57R56, 57K41 (Secondary) |
| url | https://arxiv.org/abs/2402.01081 |