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We study equivariant coarse homology theories through an axiomatic framework. To this end we introduce the category of equivariant bornological coarse spaces and construct the universal equivariant coarse homology theory with values in the…
Colored HOMFLY-PT invariant, the generalization of the colored Jones polynomial, is one of the most important quantum invariants of links. This paper is devoted to investigating the basic structures of the colored HOMFLY-PT invariants of…
We propose a framework for unifying the sl(N) Khovanov-Rozansky homology (for all N) with the knot Floer homology. We argue that this unification should be accomplished by a triply graded homology theory which categorifies the HOMFLY…
Ozsvath and Szabo gave a combinatorial description of knot Floer homology based on a cube of resolutions, which uses maps with twisted coefficients. We study the t=1 specialization of their construction. The associated spectral sequence…
It is known that the fundamental group homomorphism $\pi_1(T^2) \to \pi_1(S^3\setminus K)$ induced by the inclusion of the boundary torus into the complement of a knot $K$ in $S^3$ is a complete knot invariant. Many classical invariants of…
The aim of this paper is two-fold. First, we give a fully geometric description of the HOMFLYPT homology of Khovanov-Rozansky. Our method is to construct this invariant in terms of the cohomology of various sheaves on certain algebraic…
We show that if a positive integral surgery on a knot K inside a homology sphere X with Seifert genus g(K) results in an induced knot K_n in X_n(K)=Y which has simple Floer homology, we should have n>=2g(K). Moreover, if X is the standard…
When $\mathbf{k}$ is a field, type D structures over the algebra $\mathbf{k}[u,v]/(uv)$ are equivalent to immersed curves decorated with local systems in the twice-punctured disk. Consequently, knot Floer homology, as a type D structure…
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…
In this paper, we study the structure of the quantum cohomology ring of a projective hypersurface with non-positive 1st Chern class. We prove a theorem which suggests that the mirror transformation of the quantum cohomology of a projective…
We compute the Poincare polynomial and the cohomology algebra with rational coefficeints of the manifold M_n of real points of the moduli space of algebraic curves of genus 0 with n labeled points. This cohomology is a quadratic algebra,…
These are the notes of the lectures delivered by the author at CIME in June 2018. The main purpose of the notes is to provide an overview of the techniques used in the construction of the triply graded link homology. The homology is space…
We introduce a family of generalized Schr\"oder polynomials $S_\tau(q,t,a)$, indexed by triangular partitions $\tau$ and prove that $S_\tau(q,t,a)$ agrees with the Poincar\'e series of the triply graded Khovanov-Rozansky homology of the…
The Burau representation of braid groups and knot Floer homology share a link to the Fox calculus. We make this connection explicit, with the following outcome: if $B$ is the full Burau matrix of any braid, and $A$ is any square submatrix…
Using the duality between Wilson loop expectation values of SU(N) Chern-Simons theory on $S^3$ and topological open-string amplitudes on the local mirror of the resolved conifold, we study knots on $S^3$ and their invariants encoded in…
We study the perturbative large-$N$ expansion of the round three-sphere partition function in a class of M2-brane theories, including flavored SYM and ABJM theories as well as more general 3d theories admitting dual $(p,q)$ 5-brane web…
We prove that the HOMFLYPT polynomial of a link, colored by partitions with a fixed number of rows is a $q$-holonomic function. Specializing to the case of knots colored by a partition with a single row, it proves the existence of an…
The problem of computing the integral cohomology ring of the symmetric square of a topological space has been of interest since the 1930s, but limited progress has been made on the general case until recently. In this work we offer a…
We briefly review the current situation with various relations between knot/braid polynomials (Chern-Simons correlation functions), ordinary and extended, considered as functions of the representation and of the knot topology. These include…
The Harer-Zagier (HZ) transform maps the HOMFLY-PT polynomial into a rational function. For some special knots and links, the latter admits a simple factorised form, which is referred to as HZ factorisation. This property is preserved under…