A Steenrod square on Khovanov homology

We discuss links in thickened surfaces. We define the Khovanov–Lipshitz–Sarkar stable homotopy type and the Steenrod square for the homotopical Khovanov homology of links in thickened surfaces with genus [Formula: see text]. A surface means a closed oriented surface unless otherwise stated. Of course, a surface may or may not be the sphere. A thickened surface means a product manifold of a surface and the interval. A link in a thickened surface (respectively, a 3-manifold) means a submanifold of a thickened surface (respectively, a 3-manifold) which is diffeomorphic to a disjoint collection of circles. Our Khovanov–Lipshitz–Sarkar stable homotopy type and our Steenrod square of links in thickened surfaces with genus [Formula: see text] are stronger than the homotopical Khovanov homology of links in thickened surfaces with genus [Formula: see text]. It is the first meaningful Khovanov–Lipshitz–Sarkar stable homotopy type of links in 3-manifolds other than the 3-sphere. We point out that our theory has a different feature in the torus case.

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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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On the first Steenrod square for Chow groups

American Journal of Mathematics ◽

10.1353/ajm.2013.0003 ◽

2013 ◽

Vol 135 (1) ◽

pp. 53-63 ◽

Cited By ~ 4

Author(s):

Olivier Haution

Keyword(s):

Chow Groups ◽

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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