Related papers: On the Transverse Khovanov-Rozansky Homologies: Gr…

We show that every Lie algebroid $A$ over a manifold $P$ has a natural representation on the line bundle $Q_A = \wedge^{top}A \otimes \wedge^{top} T^*P$. The line bundle $Q_A$ may be viewed as the Lie algebroid analog of the orientation…

Let $X$ be a smooth projective curve of genus $g\geq 2$ over the complex numbers. A holomorphic triple $(E_1,E_2,\phi)$ on $X$ consists of two holomorphic vector bundles $E_{1}$ and $E_{2}$ over $X$ and a holomorphic map $\phi \colon E_{2}…

When studying deformations of an $A$-module $M$, Laudal and Yau showed that one can consider 1-cocycles in the Hochschild cohomology of $A$ with coefficients in the bi-module $End_k(M).$ With this in mind, the use of higher order Hochschild…

The first part of this paper is a short review of the construction [dg-ga/9710001] of invariants of rational homology 3-spheres and knots in terms of configuration space integrals. The second part describes the relationship between the…

A well-known conjecture states that for any $l$-component link $L$ in $S^3$, the rank of the knot Floer homology of $L$ (over any field) is less than or equal to $2^{l-1}$ times the rank of the reduced Khovanov homology of $L$. In this…

In the present paper, we construct the Khovanov homology theory for virtual links. Besides the direct approach with Z_{2} coefficients we also describe the Khovanov homology for framed links and the Khovanov homology using ``double cover''.…

In order to work with non-Nagata rings which are Nagata "up-to-completely-decomposed-universal-homeomorphism", specifically finite rank hensel valuation rings, we introduce the notions of pseudo-integral closure and pseudo-normalisation. We…

The family of negative torus links $T_{p,q}$ over a fixed number of strands $p$ admits a stable limit in reduced Khovanov homology as $q$ grows to infinity. In this paper, we endow this stable space with a bi-graded commutative algebra…

We construct analogs of Khovanov-Jacobsson classes and the Rasmussen invariant for links in the boundary of any smooth oriented 4-manifold. The main tools are skein lasagna modules based on equivariant and deformed versions of…

We construct an endomorphism of the Khovanov invariant to prove H-thinness and pairing phenomena of the invariants for alternating links. As a consequence, it follows that the Khovanov invariant of an oriented nonsplit alternating link is…

We define a homology theory of virtual links built out of the direct sum of the standard Khovanov complex with itself, motivating the name doubled Khovanov homology. We demonstrate that it can be used to show that some virtual links are…

In his famous Princeton Notes, Thurston introduced the so-called gluing equations defining the deformation variety. Later, Kashaev defined a non-commutative ring from H-triangulations of 3-manifolds and observed that for trefoil and…

This is the author's diploma thesis. We describe a simplification in the construction of Khovanov-Rozansky's categorification of quantum sl(n) link homology using the theory of maximal Cohen-Macaulay modules over hypersurface singularities…

Let $K$ be a field and let $S = K[X_1, \ldots, X_n]$. Let $I$ be a graded ideal in $S$ and let $M$ be a finitely generated graded $S$-module. We give upper bounds on the regularity of Koszul homology modules $H_i(I, M)$ for several classes…

In this paper we define an explicit basis for the $\mathfrak{gl}_n$-web algebra $H_n(\vec{k})$ (the $\mathfrak{gl}_n$ generalization of Khovanov's arc algebra) using categorified $q$-skew Howe duality. Our construction is a…

We construct a new class of scalar noncommutative multi-solitons on an arbitrary Kahler manifold by using Berezin's geometric approach to quantization and its generalization to deformation quantization. We analyze the stability condition…

We discuss a matrix of periodic holomorphic functions in the upper and lower half-plane which can be obtained from a factorization of an Andersen-Kashaev state integral of a knot complement with remarkable analytic and asymptotic properties…

For any free oriented Borel-Moore homology theory $A$, we construct an associative product on the $A$-theory of the stack of Higgs torsion sheaves over a projective curve $C$. We show that the resulting algebra $A\mathbf{Ha}_C^0$ admits a…

To every compact oriented surface that is composed entirely out of 2-dimensional 0- and 1-handles, we construct a dg category using structures arising in Khovanov homology. These dg categories form part of the 2-dimensional layer (a.k.a.…

The lattice cohomology of a reduced curve singularity is a bigraded ${\mathbb Z}[U]$-module ${\mathbb H}^*=\oplus_{q,n}{\mathbb H}^q_{2n}$, that categorifies the $\delta$-invariant and extract key geometric information from the semigroup of…