Related papers: Thin links and Conway spheres
Bordered Heegaard Floer homology is an invariant for 3-manifolds, which associates to a surface F an algebra A(Z), and to a 3-manifold Y with boundary, together with an orientation-preserving diffeomorphism \phi from F to \bdy Y, a module…
Quasi-alternating links are homologically thin for both Khovanov homology and knot Floer homology. We show that every quasi-alternating link gives rise to an infinite family of quasi-alternating links obtained by replacing a crossing with…
For each positive integer n, Khovanov and Rozansky constructed an invariant of links in the form of a doubly-graded cohomology theory whose Euler characteristic is the sl(n) link polynomial. We use Lagrangian Floer cohomology on some…
Knot Floer homology is an invariant for knots in the three-sphere for which the Euler characteristic is the Alexander-Conway polynomial of the knot. The aim of this paper is to study this homology for a class of satellite knots, so as to…
There is a generalization of Heegaard-Floer theory from ${\mathfrak{gl}}_{1|1}$ to other Lie (super)algebras $^L{\mathfrak{g}}$. The corresponding category of A-branes is solvable explicitly and categorifies quantum $U_q(^L{\mathfrak{g}})$…
We define three different types of spanning surfaces for knots in thickened surfaces. We use these to introduce new Seifert matrices, Alexander-type polynomials, genera, and a signature invariant. One of these Alexander polynomials extends…
In an earlier paper, we introduced a knot invariant for a null-homologous knot K in an oriented three-manifold Y, which is closely related to the Heegaard Floer homology of Y. In this paper we investigate some properties of these knot…
For links with vanishing pairwise linking numbers, the link components bound pairwise disjoint surfaces in $B^{4}$. In this paper, we describe the set of genera of such surfaces in terms of the $h$-function, which is a link invariant from…
In joint work with J. Rasmussen, we gave an interpretation of Heegaard Floer homology for manifolds with torus boundary in terms of immersed curves in a punctured torus. In particular, knot Floer homology is captured by this invariant.…
We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes…
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 give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary three-manifolds (with coefficients in Z/2). The descriptions are based on presenting the three-manifold as an integer surgery on a link in the…
We use Heegaard Floer homology to define an invariant of homology cobordism. This invariant is isomorphic to a summand of the reduced Heegaard Floer homology of a rational homology sphere equipped with a spin structure and is analogous to…
Building on the algebraic framework developed by Hendricks, Manolescu, and Zemke, we introduce and study a set of Floer-theoretic invariants aimed at detecting corks. Our invariants obstruct the extension of a given involution over any…
We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots. By staying within a world of topological pictures a little longer than in other…
We use bordered Floer homology to give a formula for the knot Floer homology of any (p, pn+1)-cable of a thin knot K in terms of Delta_K(t), tau(K), p, and n. We also give a formula for the Ozsvath-Szabo concordance invariant tau(K_{p,…
The Cabling Conjecture of Gonz\'alez-Acu\~na and Short holds that only cable knots admit Dehn surgery to a manifold containing an essential sphere. We approach this conjecture for thin knots using Heegaard Floer homology, primarily via…
Khovanov homology is an invariant for links in the three sphere that categorizes the Jones polynomial. We extend Khovanov's construction to links in 3-manifolds that are connected sums of orientable interval bundles over surfaces. Cutting…
In this paper we extend the idea of bordered Floer homology to knots and links in $S^3$: Using a specific Heegaard diagram, we construct gluable combinatorial invariants of tangles in $S^3$, $D^3$ and $I\times S^2$. The special case of…
For pattern knots admitting genus-one bordered Heegaard diagrams, we show the knot Floer chain complexes of the corresponding satellite knots can be computed using immersed curves. This, in particular, gives a convenient way to compute the…