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arXiv · 2312.09114

Local equivalence and refinements of Rasmussen's s-invariant

Nathan M. Dunfield·Robert Lipshitz·Dirk Schuetz
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Abstract

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.

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BibTeXRIS

Nathan M. Dunfield, Robert Lipshitz, Dirk Schuetz. 2025-11-13. Local equivalence and refinements of Rasmussen's s-invariant. https://arxiv.org/abs/2312.09114

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