scholarly journals The universal Khovanov link homology theory

2006 ◽  
Vol 6 (4) ◽  
pp. 1863-1892 ◽  
Author(s):  
Gad Naot
Keyword(s):  
Homology Theory ◽  
Link Homology

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 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 ON KHOVANOV'S COBORDISM THEORY FOR $\mathfrak{su}_3$ KNOT HOMOLOGY

2008 ◽  
Vol 17 (09) ◽  
pp. 1121-1173 ◽  
Author(s):  
SCOTT MORRISON ◽  
ARI NIEH
Keyword(s):  
Homology Theory ◽  
Detailed Calculation ◽  
Torus Knots ◽  
Link Homology ◽  
Cobordism Theory ◽  
Knot Homology ◽  
Planar Algebra

We reconsider the [Formula: see text] link homology theory defined by Knovanov in [9] and generalized by Mackaay and Vaz in [15]. With some slight modifications, we describe the theory as a map from the planar algebra of tangles to a planar algebra of complexes of "cobordisms with seams" (actually, a "canopolis"), making it local in the sense of Bar-Natan's local [Formula: see text] theory of [2]. We show that this "seamed cobordism canopolis" decategorifies to give precisely what you had both hope for and expect: Kuperberg's [Formula: see text] spider defined in [14]. We conjecture an answer to an even more interesting question about the decategorification of the Karoubi envelope of our cobordism theory. Finally, we describe how the theory is actually completely computable, and give a detailed calculation of the [Formula: see text] homology of the (2,n) torus knots.

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.

A Geometric Approach to Homology Theory

1976 ◽  
Author(s):  
S. Buonchristiano ◽  
C. P. Rourke ◽  
B. J. Sanderson
Keyword(s):  
Geometric Approach ◽  
Homology Theory

scholarly journals Blackbox computation of A ∞-algebras

2010 ◽  
Vol 17 (2) ◽  
pp. 391-404
Author(s):  
Mikael Vejdemo-Johansson
Keyword(s):  
Homology Theory ◽  
Algebra Structure ◽  
Finite Amount ◽  
Computational Work ◽  
Dg Algebra ◽  
Dg Algebras

Abstract Kadeishvili's proof of theminimality theorem [T. Kadeishvili, On the homology theory of fiber spaces, Russ. Math. Surv. 35:3 (1980), 231–238] induces an algorithm for the inductive computation of an A ∞-algebra structure on the homology of a dg-algebra. In this paper, we prove that for one class of dg-algebras, the resulting computation will generate a complete A ∞-algebra structure after a finite amount of computational work.

scholarly journals Path homology theory of edge-colored graphs

2021 ◽  
Vol 19 (1) ◽  
pp. 706-723
Author(s):  
Yuri V. Muranov ◽  
Anna Szczepkowska
Keyword(s):  
Spectral Sequence ◽  
Edge Coloring ◽  
Homology Theory ◽  
Homotopy Category ◽  
Homology Groups ◽  
Colored Graphs ◽  
Natural Filtration ◽  
Definition Of ◽  
Exact Sequences

Abstract In this paper, we introduce the category and the homotopy category of edge-colored digraphs and construct the functorial homology theory on the foundation of the path homology theory provided by Grigoryan, Muranov, and Shing-Tung Yau. We give the construction of the path homology theory for edge-colored graphs that follows immediately from the consideration of natural functor from the category of graphs to the subcategory of symmetrical digraphs. We describe the natural filtration of path homology groups of any digraph equipped with edge coloring, provide the definition of the corresponding spectral sequence, and obtain commutative diagrams and braids of exact sequences.

On the Homology Theory of Algebraic Varieties, I

1956 ◽  
Vol 63 (2) ◽  
pp. 248 ◽  
Author(s):  
Andrew H. Wallace
Keyword(s):  
Homology Theory ◽  
Algebraic Varieties
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