scholarly journals CALCULATING BAR-NATAN'S CHARACTERISTIC TWO KHOVANOV HOMOLOGY

2006 ◽  
Vol 15 (10) ◽  
pp. 1335-1356 ◽  
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.

ON CONJECTURES ABOUT POSITIVE BRAID KNOTS AND ALMOST ALTERNATING TORUS KNOTS

2010 ◽  
Vol 19 (11) ◽  
pp. 1471-1486
Author(s):  
MARKO STOŠIĆ
Keyword(s):  
Positive Integer ◽  
Homology Group ◽  
Knot Theory ◽  
Homology Theory ◽  
Khovanov Homology ◽  
Torus Knots ◽  
Link Homology

In this paper we resolve some conjectures concerning positive braid knots and almost alternating torus knots. Namely, we prove that the first Khovanov homology group of positive braid knot is trivial, as conjectured by Khovanov. Also, we generalize this result to show that the same is true in the case of Khovanov–Rozansky homology (sl(n) link homology) for any positive integer n. Moreover, by using the Khovanov homology theory, we prove the classical knot theory conjecture by Adams, that the only almost alternating torus knots are T3, 4 and T3, 5.

scholarly journals Stable homotopy refinement of quantum annular homology

2021 ◽  
Vol 157 (4) ◽  
pp. 710-769
Author(s):  
Rostislav Akhmechet ◽  
Vyacheslav Krushkal ◽  
Michael Willis
Keyword(s):  
Structural Properties ◽  
Cyclic Group ◽  
Group Action ◽  
Homology Theory ◽  
Khovanov Homology ◽  
Stable Homotopy ◽  
Spectrum Level ◽  
Link Homology ◽  
Cyclic Group Action ◽  
Equivariant Version

We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq ~2$ we associate to an annular link $L$ a naive $\mathbb {Z}/r\mathbb {Z}$ -equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $L$ as modules over $\mathbb {Z}[\mathbb {Z}/r\mathbb {Z}]$ . The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov homology. We study spectrum level lifts of structural properties of quantum annular homology.

scholarly journals OPEN-CLOSED TQFTS EXTEND KHOVANOV HOMOLOGY FROM LINKS TO TANGLES

2009 ◽  
Vol 18 (01) ◽  
pp. 87-150 ◽  
Author(s):  
AARON D. LAUDA ◽  
HENDRYK PFEIFFER
Keyword(s):  
Chain Complex ◽  
Special Kind ◽  
Homology Theory ◽  
Khovanov Homology ◽  
Quantum Field Theories ◽  
Finite Characteristic ◽  
Link Homology ◽  
Topological Quantum ◽  
Carry Over ◽  
A Chain

We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to represent a refinement of Bar-Natan's universal geometric complex algebraically, and thereby extend Khovanov homology from links to arbitrary tangles. For every plane diagram of an oriented tangle, we construct a chain complex whose terms are modules of a suitable algebra A such that there is one action of A or A op for every boundary point of the tangle. We give examples of such algebras A for which our tangle homology theory reduces to the link homology theories of Khovanov, Lee and Bar-Natan if it is evaluated for links. As a consequence of the Cardy condition, Khovanov's graded theory can only be extended to tangles if the underlying field has finite characteristic. Whenever the open-closed TQFT arises from a state-sum construction, we obtain honest planar algebra morphisms, and all composition properties of the universal geometric complex carry over to the algebraic complex. We give examples of state-sum open-closed TQFTs for which one can still determine both characteristic p Khovanov homology of links and Rasmussen's s-invariant.

scholarly journals TRIPLY-GRADED LINK HOMOLOGY AND HOCHSCHILD HOMOLOGY OF SOERGEL BIMODULES

2007 ◽  
Vol 18 (08) ◽  
pp. 869-885 ◽  
Author(s):  
MIKHAIL KHOVANOV
Keyword(s):  
Group Action ◽  
Braid Group ◽  
Polynomial Algebra ◽  
Homology Theory ◽  
Hochschild Homology ◽  
Polynomial Algebras ◽  
Homology Groups ◽  
Link Homology ◽  
Soergel Bimodules ◽  
Braid Group Action

We consider a class of bimodules over polynomial algebras which were originally introduced by Soergel in relation to the Kazhdan–Lusztig theory, and which describe a direct summand of the category of Harish–Chandra modules for sl(n). Rouquier used Soergel bimodules to construct a braid group action on the homotopy category of complexes of modules over a polynomial algebra. We apply Hochschild homology to Rouquier's complexes and produce triply-graded homology groups associated to a braid. These groups turn out to be isomorphic to the groups previously defined by Lev Rozansky and the author, which depend, up to isomorphism and overall shift, only on the closure of the braid. Consequently, our construction produces a homology theory for links.

scholarly journals On spectral sequences from Khovanov homology

2020 ◽  
Vol 20 (2) ◽  
pp. 531-564
Author(s):  
Andrew Lobb ◽  
Raphael Zentner
Keyword(s):  
Khovanov Homology ◽  
Spectral Sequences

A generalized skein relation for Khovanov homology and a categorification of the θ-invariant

2020 ◽  
pp. 1-27
Author(s):  
M. Chlouveraki ◽  
D. Goundaroulis ◽  
A. Kontogeorgis ◽  
S. Lambropoulou
Keyword(s):  
Jones Polynomial ◽  
Khovanov Homology ◽  
Spectral Sequences ◽  
Link Invariant ◽  
Type Relation ◽  
Skein Relation

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.

scholarly journals Simplicial homotopy theory, link homology and Khovanov homology

2018 ◽  
Vol 27 (07) ◽  
pp. 1841002
Author(s):  
Louis H. Kauffman
Keyword(s):  
Homotopy Theory ◽  
Khovanov Homology ◽  
Frobenius Algebra ◽  
Link Diagram ◽  
Link Homology ◽  
Simplicial Homotopy ◽  
Simplicial Methods

This paper shows how, in principle, simplicial methods, including the well-known Dold–Kan construction can be applied to convert link homology theories into homotopy theories. The paper studies particularly the case of Khovanov homology and shows how simplicial structures are implicit in the construction of the Khovanov complex from a link diagram and how the homology of the Khovanov category, with coefficients in an appropriate Frobenius algebra, is related to Khovanov homology. This Khovanov category leads to simplicial groups satisfying the Kan condition that are relevant to a homotopy theory for Khovanov homology.

scholarly journals Knot invariants arising from homological operations on Khovanov homology

2016 ◽  
Vol 25 (03) ◽  
pp. 1640012 ◽  
Author(s):  
Krzysztof K. Putyra ◽  
Alexander N. Shumakovitch
Keyword(s):  
Homology Theory ◽  
Khovanov Homology ◽  
Unified Theory ◽  
Present Evidence ◽  
Knot Invariants ◽  
Knot Invariant

We construct an algebra of nontrivial homological operations on Khovanov homology with coefficients in [Formula: see text] generated by two Bockstein operations. We use the unified Khovanov homology theory developed by the first author to lift this algebra to integral Khovanov homology. We conjecture that these two algebras are infinite and present evidence in support of our conjectures. Finally, we list examples of knots that have the same even and odd Khovanov homology, but different actions of these homological operations. This confirms that the unified theory is a finer knot invariant than the even and odd Khovanov homology combined. The case of reduced Khovanov homology is also considered.

scholarly journals Virtual Khovanov homology using cobordisms

2014 ◽  
Vol 23 (09) ◽  
pp. 1450046 ◽  
Author(s):  
Daniel Tubbenhauer
Keyword(s):  
Computer Program ◽  
Jones Polynomial ◽  
Virtual Link ◽  
Link Homology ◽  
Direct Extension ◽  
Virtual Links ◽  
Local Properties ◽  
Characteristic Two ◽  
Topological Construction

We extend Bar-Natan's cobordism-based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex (h = t = 0), the variant of Lee (h = 0, t = 1) and other classical link homologies. We show that our construction allows, over rings of characteristic two, extensions with no classical analogon, e.g. Bar-Natan's ℤ/2-link homology can be extended in two non-equivalent ways. Our construction is computable in the sense that one can write a computer program to perform calculations, e.g. we have written a MATHEMATICA-based program. Moreover, we give a classification of all unoriented TQFTs which can be used to define virtual link homologies from our topological construction. Furthermore, we prove that our extension is combinatorial and has semi-local properties. We use the semi-local properties to prove an application, i.e. we give a discussion of Lee's degeneration of virtual homology.

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