An iterable surgery formula on involutive knot lattice homotopy
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910536955854848 |
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| author | Niemi-Colvin, Seppo |
| author_facet | Niemi-Colvin, Seppo |
| contents | In ``Knots in lattice homology", Ozsváth, Stipsicz, and Szabó showed that knot lattice homology satisfies a surgery formula similar to the one relating knot Floer homology and Heegaard Floer homology, and in previous work, I showed that knot lattice homology is the persistent homology of a doubly filtered space. Here I provide an iterable version of the surgery formula that, provided the initial knot lattice space with flip map, produces a space isomorphic as a doubly-filtered space to the corresponding knot lattice space for the dual knot with the corresponding flip map. If we include the involutive data for the original knot and ambient three-manifold, we can also produce the corresponding involutive data on the new knot lattice space without assuming that the original three-manifold is an $L$-space. I construct $\infty$-categories where these operations are functorial. Finally, I use the surgery formula to compute some examples of knot lattice spaces including for the regular fiber of $Σ(2,3,7)$ and for a knot in a three-manifold that is not given by an almost rational graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_15274 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An iterable surgery formula on involutive knot lattice homotopy Niemi-Colvin, Seppo Geometric Topology 57K18 In ``Knots in lattice homology", Ozsváth, Stipsicz, and Szabó showed that knot lattice homology satisfies a surgery formula similar to the one relating knot Floer homology and Heegaard Floer homology, and in previous work, I showed that knot lattice homology is the persistent homology of a doubly filtered space. Here I provide an iterable version of the surgery formula that, provided the initial knot lattice space with flip map, produces a space isomorphic as a doubly-filtered space to the corresponding knot lattice space for the dual knot with the corresponding flip map. If we include the involutive data for the original knot and ambient three-manifold, we can also produce the corresponding involutive data on the new knot lattice space without assuming that the original three-manifold is an $L$-space. I construct $\infty$-categories where these operations are functorial. Finally, I use the surgery formula to compute some examples of knot lattice spaces including for the regular fiber of $Σ(2,3,7)$ and for a knot in a three-manifold that is not given by an almost rational graph. |
| title | An iterable surgery formula on involutive knot lattice homotopy |
| topic | Geometric Topology 57K18 |
| url | https://arxiv.org/abs/2407.15274 |