On Knot Floer Homology for Some Fibered Knots

Roberts, Lawrence P.

doi:10.1142/s0219199712500538

Cited by 1 publication

(2 citation statements)

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“…modulo the relevant natural identifications. The proof is then complete as long as one can show that the chain maps in (30) arise as the degree 0 components of a degree 0 filtered chain map as in (29). Although we do not give details, this can be arranged, defining the higher degree components of the chain map by counting instantons on the excision cobordism over higher dimensional families of metrics and perturbations, mimicking the arguments in [23,Section 6].…”

Section: Examples Of Khovanov-floer Theoriesmentioning

confidence: 91%

“…• study the knot Floer homology of fibered knots [31,30], • establish tightness and non-fillability of certain contact structures [3], • prove that Khovanov homology detects the unknot [23], • prove that Khovanov's categorification of the n-colored Jones polynomial detects the unknot for n ≥ 2 [13], • detect the unknot with Khovanov homology of certain satellites [15,17], • prove that Khovanov homology detects the unlink [16,5].…”

Section: Introductionmentioning

confidence: 99%

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On the functoriality of Khovanov–Floer theories

Baldwin

Hedden

Lobb

2019

Advances in Mathematics

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We introduce the notion of a Khovanov-Floer theory. Roughly, such a theory assigns a filtered chain complex over Z/2Z 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 S 3 × [0, 1] between links in S 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 Szabó'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.

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Section: Examples Of Khovanov-floer Theoriesmentioning

confidence: 91%