Related papers: Geometric proof of Thom conjecture
A theorem of Kronheimer and Mrowka states that Khovanov homology is able to detect the unknot. That is, if a knot has the Khovanov homology of the unknot, then it is equivalent to it. Similar results hold for the trefoils and the…
This paper has been withdrawn by author due to an error in the proof.
The purpose of this paper is to construct and study equivariant Khovanov homology - a version of Khovanov homology theory for periodic links. Since our construction works regardless of the characteristic of the coefficient ring it…
This paper has been withdrawn by the author due to a mistake in the proof of the main theorem.
We prove that Khovanov homology with coefficients in $\mathbb{Z}/2\mathbb{Z}$ detects the $(2,5)$ torus knot. Our proof makes use of a wide range of deep tools in Floer homology, Khovanov homology, and Khovanov homotopy. We combine these…
Two conjectures recently proposed by one of the authors are disproved
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…
This is an expository paper discussing various versions of Khovanov homology theories, interrelations between them, their properties, and their applications to other areas of knot theory and low-dimensional topology.
In this paper we correct an inaccuracy that appears in the proof of Theorem 1. in Czerwik's article "Contraction mappings in $b$-metric spaces.", Acta Math. Inform. Univ. Ostraviensis, 1:5--11, 1993.
Inspired by bordered Floer homology, we describe a type A structure on a Khovanov homology for a tangle, which complements the type D structure in a previous paper. The type A structure is a differential module over a certain algebra. This…
We prove a conjecture of Barraud and Cornea in the monotone setting, refining a result of Chekanov on the Hofer distance between two Hamiltonian isotopic Lagrangian submanifolds.
In 1998 D. Tamarkin announced a proof of Kontsevich formality theorem based on the existence of structure of homotopy Gerstenhaber algebra in the Hochschild cochains of an associative algebra. In this note we give a detailed explanation of…
We give a simple proof of Lee's result from [Adv. Math. 179 (2005) 554-586; arXiv:math.GT/0210213], that the dimension of the Lee variant of the Khovanov homology of a c-component link is 2^c, regardless of the number of crossings. Our…
This paper has been withdrawn because of serious errors.
We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that…
We study the twisted Novikov homology of the complement of a complex hypersurface in general position at infinity. We give a self-contained topological proof of the vanishing (except possibly in the middle degree) of the twisted Novikov…
We determine the algebraic structure underlying the geometric complex associated to a link in Bar-Natan's geometric formalism of Khovanov's link homology theory (n=2). We find an isomorphism of complexes which reduces the complex to one in…
Many important problems in extremal combinatorics can be be stated as proving a pure binomial inequality in graph homomorphism numbers, i.e., proving that…
We prove that every $\mathbb{Z}_2$H-thin link has no $2^k$-torsion for $k>1$ in its Khovanov homology. Together with previous results by Eun Soo Lee and the author, this implies that integer Khovanov homology of non-split alternating links…
We enhance the Khovanov TQFT using basepoint actions, over the field with two elements. Our enhanced Khovanov TQFT behaves similarly to gauge/Floer theoretic invariants of the double branched cover with opposite orientation: they both are…