Local equivalence and refinements of Rasmussen's s-invariant
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866911263926255616 |
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| author | Dunfield, Nathan M. Lipshitz, Robert Schuetz, Dirk |
| author_facet | Dunfield, Nathan M. Lipshitz, Robert Schuetz, Dirk |
| contents | Inspired by the notions of local equivalence in monopole and Heegaard Floer homology, we introduce a version of local equivalence that combines odd Khovanov homology with equivariant even Khovanov homology into an algebraic package called a local even-odd (LEO) triple. We get a homomorphism from the smooth concordance group $C$ to the resulting local equivalence group $C_{LEO}$ of such triples. We give several versions of the $s$-invariant that descend to $C_{LEO}$, including one that completely determines whether the image of a knot $K$ in $C_{LEO}$ is trivial. We discuss computer experiments illustrating the power of these invariants in obstructing sliceness, both statistically and for some interesting knots studied by Manolescu-Piccirillo. Along the way, we explore several variants of this local equivalence group, including one that is totally ordered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_09114 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Local equivalence and refinements of Rasmussen's s-invariant Dunfield, Nathan M. Lipshitz, Robert Schuetz, Dirk Geometric Topology 57K18 (Primary) 57K10, 57-08 (Secondary) Inspired by the notions of local equivalence in monopole and Heegaard Floer homology, we introduce a version of local equivalence that combines odd Khovanov homology with equivariant even Khovanov homology into an algebraic package called a local even-odd (LEO) triple. We get a homomorphism from the smooth concordance group $C$ to the resulting local equivalence group $C_{LEO}$ of such triples. We give several versions of the $s$-invariant that descend to $C_{LEO}$, including one that completely determines whether the image of a knot $K$ in $C_{LEO}$ is trivial. We discuss computer experiments illustrating the power of these invariants in obstructing sliceness, both statistically and for some interesting knots studied by Manolescu-Piccirillo. Along the way, we explore several variants of this local equivalence group, including one that is totally ordered. |
| title | Local equivalence and refinements of Rasmussen's s-invariant |
| topic | Geometric Topology 57K18 (Primary) 57K10, 57-08 (Secondary) |
| url | https://arxiv.org/abs/2312.09114 |