Related papers: The Superpolynomial for Knot Homologies

We prove for the first time that knot Floer homology and Khovanov homology can detect non-fibered knots, and that HOMFLY homology detects infinitely many such knots; these theories were previously known to detect a mere six knots, all…

We show that the triply graded Khovanov-Rozansky homology of knots and links over a field of positive odd characteristic $p$ descends to an invariant in the homotopy category finite-dimensional $p$-complexes. A $p$-extended differential on…

We give new link detection results for knot and link Floer homology inspired by recent work on Khovanov homology. We show that knot Floer homology detects $T(2,4)$, $T(2,6)$, $T(3,3)$, $L7n1$, and the link $T(2,2n)$ with the orientation of…

We define a homology $\mathcal{H}_N$ for closed braids by applying Khovanov and Rozansky's matrix factorization construction with potential $ax^{N+1}$. Up to a grading shift, $\mathcal{H}_0$ is the HOMFLYPT homology defined in…

We establish an isomorphism between the Khovanov-Rozansky triply graded link homology and the geometric triply graded homology due to the authors. Hence we provide an interpretation of the Khovanov-Rozansky homology of the closure of a…

Cobordisms are naturally bigraded and we show that this grading extends to Khovanov homology, making it a triply graded theory. Although the new grading does not make the homology a stronger invariant, it can be used to show that odd…

We review the construction of Heegaard Floer homology for closed three-manifolds and also for knots and links in the three-sphere. We also discuss three applications of this invariant to knot theory: studying the Thurston norm of a link…

In this paper, we introduce a sequence of invariants of a knot K in S^3: the knot Floer homology groups of the preimage of K in the m-fold cyclic branched cover over K. We exhibit the knot Floer homology in the m-fold branched cover as the…

The stable Khovanov-Rozansky homology of torus knots has been conjecturally described as the Koszul homology of an explicit non-regular sequence of polynomials. We verify this conjecture against newly available computational data for…

We introduce an invariant of tangles in Khovanov homology by considering a natural inverse system of Khovanov homology groups. As application, we derive an invariant of strongly invertible knots; this invariant takes the form of a graded…

We define and study a family of link invariants $\mathit{HFK}_{n}(L)$. Although these homology theories are defined using holomorphic disc counts, they share many properties with $sl_{n}$ homology. Using these theories, we give a framework…

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 these notes, I will sketch a new approach to Khovanov homology of knots and links based on counting the solutions of certain elliptic partial differential equations in four and five dimensions. The equations are formulated on four and…

Let $E_{k}^{F}(D)$ be the spectral sequence induced by the oriented cube of resolutions on knot Floer homology. We prove that $E_{2}^{F}(D)$ is a triply graded link invariant whose graded Euler characteristic is the HOMFLY-PT polynomial and…

We follow the same technics we used before in \cite{AZ} of extending knot Floer homology to embedded graphs in a 3-manifold, by using the Kauffman topological invariant of embedded graphs by associating family of links and knots to a such…

It is known that knot Floer homology detects the genus and Alexander polynomial of a knot. We investigate whether knot Floer homology of $K$ detects more structure of minimal genus Seifert surfaces for $K$. We define an invariant of…

Using the theory of perverse sheaves of vanishing cycles, we define a homological invariant of knots in three-manifolds, similar to the three-manifold invariant constructed by Abouzaid and the second author. We use spaces of SL(2,C) flat…

We use categorical annular evaluation to give a uniform construction of both $\mathfrak{sl}_n$ and HOMFLYPT Khovanov-Rozansky link homology, as well as annular versions of these theories. Variations on our construction yield…

We compute the reduced version of Khovanov and Rozansky's sl(N) homology for two-bridge knots and links. The answer is expressed in terms of the HOMFLY polynomial and signature.

In this thesis we work with Khovanov homology of links and its generalizations, as well as with the homology of graphs. Khovanov homology of links consists of graded chain complexes which are link invariants, up to chain homotopy, with…