Related papers: Khovanov homology and Rasmussen's s-invariants for…
We conjecture a relation between the sl(N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to new regularities among the sl(N) knot…
We evaluate the differences of HOMFLY-PT invariants for pairs of mutant knots colored with representations of $SL(N)$, which are large enough to distinguish between them. These mutant pairs include the pretzel mutants, which require at…
We define two concordance invariants of knots using framed instanton homology. These invariants $\nu^\sharp$ and $\tau^\sharp$ provide bounds on slice genus and maximum self-linking number, and the latter is a concordance homomorphism which…
We prove that the hypothetical extreme Khovanov cohomology of a link is the cohomology of the independence simplicial complex of its Lando graph. We also provide a family of knots having as many non-trivial extreme Khovanov cohomology…
We present an easy example of mutant links with different Khovanov homology. The existence of such an example is important because it shows that Khovanov homology cannot be defined with a skein rule similar to the skein relation for the…
Kanenobu has given infinite families of knots with the same HOMFLY polynomials. We show that these knots also have the same sl(n) and HOMFLY homologies, thus giving the first example of an infinite family of knots undistinguishable by these…
We study the relationship between Bar-Natan's perturbation in Khovanov homology and Szabo's geometric spectral sequence, and construct a link invariant that generalizes both into a common theory. We study a few properties of the new…
We collect statistics which consist of the coefficients in the expansion of the generating polynomials that count the Kauffman states associated with certain classes of pretzel knots having n tangles, of r half-twists respectively.
Given a diagram D of a knot K, we give easily computable bounds for Rasmussen's concordance invariant s(K). The bounds are not independent of the diagram D chosen, but we show that for diagrams satisfying a given condition the bounds are…
We establish a direct map between refined topological vertex and sl(N) homological invariants of the of Hopf link, which include Khovanov-Rozansky homology as a special case. This relation provides an exact answer for homological invariants…
Building on work by Alishahi-Dowlin, we extract a new knot invariant $\lambda \ge 0$ from universal Khovanov homology. While $\lambda$ is a lower bound for the unknotting number, in fact more is true: $\lambda$ is a lower bound for the…
We prove that Khovanov homology detects the trefoils. Our proof incorporates an array of ideas in Floer homology and contact geometry. It uses open books; the contact invariants we defined in the instanton Floer setting; a bypass exact…
Khovanov homology is a powerful link invariant: a categorification of the Jones polynomial that enjoys a rich and beautiful algebraic structure. This homology theory has been extensively studied and it has become an ubiquitous topic in…
Computing polynomial invariants for knots and links using braid representations relies heavily on finding the trace of Hecke algebra elements. There is no easy method known for computing the trace and hence it becomes difficult to compute…
We determine a wide class of knots, which includes unknotting number one knots, within which Khovanov homology detects the unknot. A corollary is that the Khovanov homology of many satellite knots, including the Whitehead double, detects…
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…
This article provides an overview of relative strengths of polynomial invariants of knots and links, such as the Alexander, Jones, Homflypt, Kaufman two-variable polynomial, and Khovanov polynomial.
We prove that the Khovanov homology of alternating knots and 2-component links is equal (as a singly graded group) to the singular homology of a certain space of trace- free, binary dihedral representations of the link group. More…
We construct the concordance invariant coming from the $E(-1)$ spectral sequence on Khovanov homology in the same way Rasmussen's $s$ invariant comes from the Lee spectral sequence, and show that it gives a bound on the nonorientable slice…
We present a combinatorial method for a calculation of knot Floer homology with Z-coefficient of (1,1)-knots, and then demonstrate it for non-alternating (1,1)-knots with ten crossings and the pretzel knots of type (-2,m,n). Our…