Related papers: A refinement of Khovanov homology
We show that the unnormalised Khovanov homology of an oriented link can be identified with the derived functors of the inverse limit. This leads to a homotopy theoretic interpretation of Khovanov homology.
Khovanov homology is a recently introduced invariant of oriented links in $\mathbb{R}^3$. It categorifies the Jones polynomial in the sense that the (graded) Euler characteristic of the Khovanov homology is a version of the Jones polynomial…
We lift the characteristic-2 totally twisted Khovanov homology of Roberts and Jaeger to a theory with integer coefficients. The result is a complex computing reduced odd Khovanov homology for knots. This complex is equivalent to a…
We give a simple recursion which computes the triply graded Khovanov-Rozansky homology of several infinite families of knots and links, including the $(n,nm\pm 1)$ and $(n,nm)$ torus links for $n,m\geq 1$. We interpret our results in terms…
In this paper, we describe a canopolis (i.e. categorified planar algebra) formalism for Khovanov and Rozansky's link homology theory. We show how this allows us to organize simplifications in the matrix factorizations appearing in their…
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…
We show that Rasmussen's invariant of knots, which is derived from Lee's variant of Khovanov homology, is equal to an analogous invariant derived from certain other filtered link homologies.
We apply the techniques of totally twisted Khovanov homology to the constructions by M. Asaeda, J. Przytycki, and A. Sikora of Khovanov type homologies for links and tangles in I-bundles over (orientable) surfaces. As a result we describe…
In a previous paper we constructed a spectrum-level refinement of Khovanov homology. This refinement induces stable cohomology operations on Khovanov homology. In this paper we show that these cohomology operations commute with cobordism…
Let L be a link in an thickened annulus. We specify the embedding of this annulus in the three sphere, and consider its complement thought of as the axis to L. In the right circumstances this axis lifts to a null-homologous knot in the…
We construct explicitly the Khovanov homology theory for virtual links with arbitrary coefficients by using the twisted coefficients method. This method also works for constructing Khovanov homology for ``non-oriented virtual knots'' in the…
The purpose of this paper is to interpret polynomial invariants of strongly invertible links in terms of Khovanov homology theory. To a divide, that is a proper generic immersion of a finite number of copies of the unit interval and circles…
In this note, we prove the existence of a tri-graded Khovanov-type bicomplex (Theorem 1.2). The graded Euler characteristic of the total complex associated with this bicomplex is the colored Jones polynomial of a link. The first grading of…
Examples of knots and links distinguished by the total rank of their Khovanov homology but sharing the same two-fold branched cover are given. As a result, Khovanov homology does not yield an invariant of two-fold branched covers.
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…
We investigate Bar-Natan's characteristic two Khovanov link homology theory studying both the filtered and bi-graded theories. The filtered theory is computed explicitly and the bi-graded theory analysed by setting up a family of spectral…
From Khovanov homology, we extract a new lower bound for the Gordian distance of knots, which combines and strengthens the previously existing bounds coming from Rasmussen invariants and from torsion invariants. We also improve the bounds…
We define an invariant of transverse links in the standard contact 3-sphere as a distinguished element of the Khovanov homology of the link. The quantum grading of this invariant is the self-linking number of the link. For knots, this gives…
We investigate several conjectures in geometric topology by assembling computer data obtained by studying weaving knots, a doubly infinite family $W(p,n)$ of examples of hyperbolic knots. In particular, we compute some important polynomial…
We give a new, elementary proof that Khovanov homology with $\mathbb{Z}/2\mathbb{Z}$--coefficients is invariant under Conway mutation. This proof also gives a strategy to prove Baldwin and Levine's conjecture that $\delta$--graded knot…