scholarly journals Instantons and Bar-Natan homology

2021 ◽  
Vol 157 (3) ◽  
pp. 484-528
Author(s):  
P. B. Kronheimer ◽  
T. S. Mrowka
Keyword(s):  
Spectral Sequence ◽  
Khovanov Homology ◽  
Characteristic 2 ◽  
Knots And Links

A spectral sequence is established whose $E_{2}$ page is Bar-Natan's variant of Khovanov homology and which abuts to a deformation of instanton homology for knots and links. This spectral sequence arises as a specialization of a spectral sequence whose $E_{2}$ page is a characteristic-2 version of $F_{5}$ homology in Khovanov's classification.

scholarly journals Curved Rickard complexes and link homologies

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).

scholarly journals A VOLUME FORM ON THE KHOVANOV INVARIANT

2012 ◽  
Vol 21 (04) ◽  
pp. 1250032
Author(s):  
JUAN ORTIZ-NAVARRO
Keyword(s):  
Chain Complex ◽  
Khovanov Homology ◽  
Volume Form ◽  
Reidemeister Torsion ◽  
Homology Groups ◽  
Numerical Invariant ◽  
Reidemeister Moves ◽  
Invariant Volume ◽  
Knots And Links

The Reidemeister torsion construction can be applied to the chain complex used to compute the Khovanov homology of a knot or a link. This defines a volume form on Khovanov homology. The volume form transforms correctly under Reidemeister moves to give an invariant volume on the Khovanov homology. In this paper, its construction and invariance under these moves is demonstrated. Also, some examples of the invariant are presented for particular choices for the bases of homology groups to obtain a numerical invariant of knots and links. In these examples, the algebraic torsion seen in the Khovanov chain complex when homology is computed over ℤ is recovered.

scholarly journals A link-splitting spectral sequence in Khovanov homology

2015 ◽  
Vol 164 (5) ◽  
pp. 801-841 ◽  
Author(s):  
Joshua Batson ◽  
Cotton Seed
Keyword(s):  
Spectral Sequence ◽  
Khovanov Homology

scholarly journals ON THE QUANTUM FILTRATION OF THE UNIVERSAL sl(2) FOAM COHOMOLOGY

2012 ◽  
Vol 21 (03) ◽  
pp. 1250023 ◽  
Author(s):  
CARMEN CAPRAU
Keyword(s):  
Lower Bound ◽  
Spectral Sequence ◽  
Khovanov Homology ◽  
Reidemeister Moves

We investigate the filtered theory corresponding to the universal sl(2) foam cohomology [Formula: see text] for links, where a, h ∈ ℂ. We show that there is a spectral sequence converging to [Formula: see text] which is invariant under the Reidemeister moves, and whose E1 term is isomorphic to Khovanov homology. This spectral sequence can be used to obtain from the foam perspective an analogue of the Rasmussen invariant and a lower bound for the slice genus of a knot.

scholarly journals Khovanov homology, Lee homology and a Rasmussen invariant for virtual knots

2017 ◽  
Vol 26 (03) ◽  
pp. 1741001 ◽  
Author(s):  
Heather A. Dye ◽  
Aaron Kaestner ◽  
Louis H. Kauffman
Keyword(s):  
Large Class ◽  
Jones Polynomial ◽  
Khovanov Homology ◽  
Frobenius Algebra ◽  
Conjugation Operator ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Link Diagrams ◽  
Knots And Links

The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov–Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly applies as well to classical knots and classical knot cobordisms. We give an alternate formulation for the Manturov definition [34] of Khovanov homology [25], [26] for virtual knots and links with arbitrary coefficients. This approach uses cut loci on the knot diagram to induce a conjugation operator in the Frobenius algebra. We use this to show that a large class of virtual knots with unit Jones polynomial is non-classical, proving a conjecture in [20] and [10]. We then discuss the implications of the maps induced in the aforementioned theory to the universal Frobenius algebra [27] for virtual knots. Next we show how one can apply the Karoubi envelope approach of Bar-Natan and Morrison [3] on abstract link diagrams [17] with cross cuts to construct the canonical generators of the Khovanov–Lee homology [30]. Using these canonical generators we derive a generalization of the Rasmussen invariant [39] for virtual knot cobordisms and generalize Rasmussen’s result on the slice genus for positive knots to the case of positive virtual knots. It should also be noted that this generalization of the Rasmussen invariant provides an easy to compute obstruction to knot cobordisms in [Formula: see text] in the sense of Turaev [42].

scholarly journals Spectral sequences for commutative Lie algebras

2020 ◽  
Vol 28 (2) ◽  
pp. 123-137
Author(s):  
Friedrich Wagemann
Keyword(s):  
Spectral Sequence ◽  
Lie Algebras ◽  
Spectral Sequences ◽  
Characteristic 2 ◽  
Leibniz Cohomology

AbstractWe construct some spectral sequences as tools for computing commutative cohomology of commutative Lie algebras in characteristic 2. In a first part, we focus on a Hochschild-Serre-type spectral sequence, while in a second part we obtain spectral sequences which compare Chevalley--Eilenberg-, commutative- and Leibniz cohomology. These methods are illustrated by a few computations.

scholarly journals Khovanov homology and knot Floer homology for pointed links

2017 ◽  
Vol 26 (02) ◽  
pp. 1740004 ◽  
Author(s):  
John A. Baldwin ◽  
Adam Simon Levine ◽  
Sucharit Sarkar
Keyword(s):  
Spectral Sequence ◽  
Floer Homology ◽  
Khovanov Homology ◽  
Knot Floer Homology ◽  
Text Page ◽  
Text Component

A well-known conjecture states that for any [Formula: see text]-component link [Formula: see text] in [Formula: see text], the rank of the knot Floer homology of [Formula: see text] (over any field) is less than or equal to [Formula: see text] times the rank of the reduced Khovanov homology of [Formula: see text]. In this paper, we describe a framework that might be used to prove this conjecture. We construct a modified version of Khovanov homology for links with multiple basepoints and show that it mimics the behavior of knot Floer homology. We also introduce a new spectral sequence converging to knot Floer homology whose [Formula: see text] page is conjecturally isomorphic to our new version of Khovanov homology; this would prove that the conjecture stated above holds over the field [Formula: see text].

scholarly journals KHOVANOV HOMOLOGY AND RASMUSSEN'S s-INVARIANTS FOR PRETZEL KNOTS

2010 ◽  
Vol 19 (09) ◽  
pp. 1183-1204 ◽  
Author(s):  
RYOHEI SUZUKI
Keyword(s):  
Spectral Sequence ◽  
Khovanov Homology ◽  
Pretzel Knots ◽  
Almost All

We calculate the rational Khovanov homology of a class of pretzel knots by using the spectral sequence constructed by Turner. Moreover, we determine Rasmussen's s-invariant of almost all pretzel knots P(p, q, r) by using Turner's spectral sequence, a sharper slice-Bennequin inequality, and a skein inequality.

Sign in / Sign up

Export Citation Format

Share Document