Related papers: Categorification and applications in topology and …
We use the knot homology of Khovanov and Lee to construct link concordance invariants generalizing the Rasmussen $s$-invariant of knots. The relevant invariant for a link is a filtration on a vector space of dimension $2^{|L|}$. The basic…
This paper formulates a generalization of our work on quantum knots to explain how to make quantum versions of algebraic, combinatorial and topological structures. We include a description of previous work on the construction of Hilbert…
This is a draft of a book submitted for publication by the AMS. Its theme is the remarkable interplay, accelerating in the last few decades, between topology and the theory of orderable groups, with applications in both directions. It…
Motivated by physical constructions of homological knot invariants, we study their analogs for closed 3-manifolds. We show that fivebrane compactifications provide a universal description of various old and new homological invariants of…
By analogy with algebraic geometry, we define a category of non-linear sheaves (quasi-coherent homotopy-sheaves of topological spaces) on projective toric varieties and prove a splitting result for its algebraic K-theory, generalising…
The inclusion of the unit in a braided tensor category $\mathcal{V}$ induces a 1-morphism in the Morita 4-category of braided tensor categories $BrTens$. We give criteria for the dualizability of this morphism. When $\mathcal{V}$ is a…
The Khovanov-Rozansky (KR) link polynomial is a certain $t$-deformation of Wilson loops in 3-dimensional $SU(N)$ Chern--Simons topological field theory, believed to be an observable in the refined Chern-Simons theory, probably described in…
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''.…
We study algebraic varieties parametrized by topological spaces and enlarge the domains of Lawson homology and morphic cohomology to this category. We prove a Lawson suspension theorem and splitting theorem. A version of Friedlander-Lawson…
We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to extend Khovanov homology from links to arbitrary tangles, not necessarily even. For every plane diagram of…
This paper contains a categorification of the sl(k) link invariant using parabolic singular blocks of category O. Our approach is intended to be as elementary as possible, providing combinatorial proofs of the main results of Sussan. We…
In the first section we discuss Morita invariance of differentiable/algebroid cohomology. In the second section we present an extension of the van Est isomorphism to groupoids. This immediately implies a version of Haefliger's conjecture…
Frobenius extensions play a central role in the link homology theories based upon the sl(n) link variants, and each of these Frobenius extensions may be recast geometrically via a category of marked cobordisms in the manner of Bar-Natan.…
We construct an infinite family of homology theories of framed links in thickened surfaces, as well as a homology theory whose graded Euler characteristic is exactly the Kauffman bracket of the link in the surface. Both theories are based…
Khovanov-Lauda define a 2-category $\mathcal{U}$ such that the split Grothendieck group $K_0(\mathcal{U})$ is isomorphic to an integral version of the quantized universal enveloping algebra $\mathbf{U}(\mathfrak{sl}_n)$, $n \geq 2$.…
Kontsevich and Soibelman reformulated and slightly generalised the topological recursion of math-ph/0702045, seeing it as a quantization of certain quadratic Lagrangians in $T^*V$ for some vector space $V$. KS topological recursion is a…
In this paper we compute Lawson homology groups and semi-topological K-theory for some threefolds and fourfolds. We consider smooth complex projective varieties whose zero cycles are supported on a proper subvariety. Rationally connected…
We introduce a method for associating a chain complex to a module over a combinatorial category, such that if the complex is exact then the module has a rational Hilbert series. We prove homology--vanishing theorems for these complexes for…
We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various…
We begin the study of categorifications of Donaldson-Thomas invariants associated with Hilbert schemes of points on the three-dimensional affine space, which we call DT categories. The DT category is defined to be the category of matrix…