On spectral sequences from Khovanov homology

The Jones polynomial is a famous link invariant that can be defined diagrammatically via a skein relation. Khovanov homology is a richer link invariant that categorifies the Jones polynomial. Using spectral sequences, we obtain a skein-type relation satisfied by the Khovanov homology. Thanks to this relation, we are able to generalize the Khovanov homology in order to obtain a categorification of the θ-invariant, which is itself a generalization of the Jones polynomial.

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CALCULATING BAR-NATAN'S CHARACTERISTIC TWO KHOVANOV HOMOLOGY

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216506005111 ◽

2006 ◽

Vol 15 (10) ◽

pp. 1335-1356 ◽

Cited By ~ 18

Author(s):

PAUL R. TURNER

Keyword(s):

Homology Theory ◽

Khovanov Homology ◽

Spectral Sequences ◽

Link Homology ◽

Characteristic Two

We investigate Bar-Natan's characteristic two Khovanov link homology theory studying both the filtered and bi-graded theories. The filtered theory is computed explicitly and the bi-graded theory analysed by setting up a family of spectral sequences. The E2-pages can be described in terms of groups arising from the action of a certain endomorphism on 𝔽2-Khovanov homology. Some simple consequences are discussed.

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Curved Rickard complexes and link homologies

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2019-0044 ◽

2020 ◽

Vol 2020 (769) ◽

pp. 87-119

Author(s):

Sabin Cautis ◽

Aaron D. Lauda ◽

Joshua Sussan

Keyword(s):

Spectral Sequence ◽

Quantum Groups ◽

Braid Group ◽

Derived Categories ◽

Duke Math ◽

Khovanov Homology ◽

Hilbert Schemes ◽

Coherent Sheaves ◽

Knot Homology ◽

Original Motivation

AbstractRickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y-ification of Khovanov–Rozansky homology, preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves IV, colored links, Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).

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Counterexamples to Hochschild-Kostant-Rosenberg in characteristic p

Forum of Mathematics Sigma ◽

10.1017/fms.2021.20 ◽

2021 ◽

Vol 9 ◽

Author(s):

Benjamin Antieau ◽

Bhargav Bhatt ◽

Akhil Mathew

Keyword(s):

Spectral Sequence ◽

Spectral Sequences ◽

Characteristic P ◽

De Rham

Abstract We give counterexamples to the degeneration of the Hochschild-Kostant-Rosenberg spectral sequence in characteristic p, both in the untwisted and twisted settings. We also prove that the de Rham-HP and crystalline-TP spectral sequences need not degenerate.

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A Steenrod square on Khovanov homology

Journal of Topology ◽

10.1112/jtopol/jtu005 ◽

2014 ◽

Vol 7 (3) ◽

pp. 817-848 ◽

Cited By ~ 12

Author(s):

Robert Lipshitz ◽

Sucharit Sarkar

Keyword(s):

Khovanov Homology ◽

Steenrod Square

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On independence of iterated Whitehead doubles in the knot concordance group

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216518500037 ◽

2018 ◽

Vol 27 (01) ◽

pp. 1850003

Author(s):

Kyungbae Park

Keyword(s):

Correction Term ◽

Khovanov Homology ◽

Floer Theory ◽

Knot Concordance ◽

Torus Knot ◽

Linearly Independent ◽

Heegaard Floer

Let [Formula: see text] be the positively clasped untwisted Whitehead double of a knot [Formula: see text], and [Formula: see text] be the [Formula: see text] torus knot. We show that [Formula: see text] and [Formula: see text] are linearly independent in the smooth knot concordance group [Formula: see text] for each [Formula: see text]. Further, [Formula: see text] and [Formula: see text] generate a [Formula: see text] summand in the subgroup of [Formula: see text] generated by topologically slice knots. We use the concordance invariant [Formula: see text] of Manolescu and Owens, using Heegaard Floer correction term. Interestingly, these results are not easily shown using other concordance invariants such as the [Formula: see text]-invariant of knot Floer theory and the [Formula: see text]-invariant of Khovanov homology. We also determine the infinity version of the knot Floer complex of [Formula: see text] for any [Formula: see text] generalizing a result for [Formula: see text] of Hedden, Kim and Livingston.

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