On the functoriality of Khovanov-Floer theories
| Autor: | Andrew Lobb, John A. Baldwin, Matthew Hedden |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Khovanov homology
Pure mathematics General Mathematics 010102 general mathematics Geometric Topology (math.GT) Cobordism Homology (mathematics) 16. Peace & justice 01 natural sciences Mathematics::Algebraic Topology Mathematics::Geometric Topology Mathematics - Geometric Topology Floer homology 57M27 Mathematics::K-Theory and Homology 0103 physical sciences Spectral sequence FOS: Mathematics Isotopy 010307 mathematical physics 0101 mathematics Invariant (mathematics) Mathematics::Symplectic Geometry Knot (mathematics) Mathematics |
| Zdroj: | Advances in mathematics, 2019, Vol.345, pp.1162-1205 [Peer Reviewed Journal] |
| Popis: | We introduce the notion of a Khovanov-Floer theory. Roughly, such a theory assigns a filtered chain complex over Z/2 to a link diagram such that (1) the E_2 page of the resulting spectral sequence is naturally isomorphic to the Khovanov homology of the link; (2) this filtered complex behaves nicely under planar isotopy, disjoint union, and 1-handle addition; and (3) the spectral sequence collapses at the E_2 page for any diagram of the unlink. We prove that a Khovanov-Floer theory naturally yields a functor from the link cobordism category to the category of spectral sequences. In particular, every page (after E_1) of the spectral sequence accompanying a Khovanov-Floer theory is a link invariant, and an oriented cobordism in R^3 \times [0,1] between links in R^3 induces a map between each page of their spectral sequences, invariant up to smooth isotopy of the cobordism rel boundary. We then show that the spectral sequences relating Khovanov homology to Heegaard Floer homology and singular instanton knot homology are induced by Khovanov-Floer theories and are therefore functorial in the manner described above, as has been conjectured for some time. We further show that Szabo's geometric spectral sequence comes from a Khovanov-Floer theory, and is thus functorial as well. In addition, we illustrate how our framework can be used to give another proof that Lee's spectral sequence is functorial and that Rasmussen's invariant is a knot invariant. Finally, we use this machinery to define some potentially new knot invariants. 42 pages. v2: 43 pages, very slight rewrite, role of basepoints treated in more detail |
| Databáze: | OpenAIRE |
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