3-manifold invariants and periodicity of homology spheres.
A 2-category of chronological cobordisms and odd Khovanov homology
We create a framework for odd Khovanov homology in the spirit of Bar-Natan's construction for the ordinary Khovanov homology. Namely, we express the cube of resolutions of a link diagram as a diagram in a certain 2-category of chronological cobordisms and show that it is 2-commutative: the composition of 2-morphisms along any 3-dimensional subcube is trivial. This allows us to create a chain complex whose homotopy type modulo certain relations is a link invariant. Both the original and the odd Khovanov...
A categorification for the chromatic polynomial.
A cohomology theory for colored tangles
We employ the sl(2) foam cohomology to define a cohomology theory for oriented framed tangles whose components are labeled by irreducible representations of . We show that the corresponding colored invariants of tangles can be assembled into invariants of bigger tangles. For the case of knots and links, the corresponding theory is a categorification of the colored Jones polynomial, and provides a tool for efficient computations of the resulting colored invariant of knots and links. Our theory is...
A colored Khovanov bicomplex
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 the bicomplex is a homological one derived from cabling of the link (i.e., replacing a strand of the link by several parallel strands); the second grading is related to the homological grading of ordinary Khovanov homology; finally, the third grading is preserved...
A colored 𝔰𝔩(N) homology for links in S³ [Book]
A combinatorial formula for the signature of alternating diagrams
We express the signature of an alternating link in terms of some combinatorial characteristics of its diagram.
A computation in Khovanov-Rozansky homology
We investigate the Khovanov-Rozansky invariant of a certain tangle and its compositions. Surprisingly the complexes we encounter reduce to ones that are very simple. Furthermore, we discuss a "local" algorithm for computing Khovanov-Rozansky homology and compare our results with those for the "foam" version of sl₃-homology.
A computation of the Kontsevich integral of torus knots.
A conjecture on Khovanov's invariants
We formulate a conjectural formula for Khovanov's invariants of alternating knots in terms of the Jones polynomial and the signature of the knot.
A Euclidean geometric invariant of framed (Un)Knots in manifolds.
A functor-valued invariant of tangles.
A geometric interpretation of Milnor's triple linking numbers.
A Jones polynomial for braid-like isotopies of oriented links and its categorification.
A lattice of finite-type invariants of virtual knots
We construct an infinite commutative lattice of groups whose dual spaces give Kauffman finite-type invariants of long virtual knots. The lattice is based "horizontally" upon the Polyak algebra and extended "vertically" using Manturov's functorial map f. For each n, the n-th vertical line in the lattice contains an infinite-dimensional subspace of Kauffman finite-type invariants of degree n. Moreover, the lattice contains infinitely many inequivalent extensions of the Conway polynomial to long virtual...
A Legendrian Thurston-Bennequin bound from Khovanov homology.
A new invariant on hyperbolic Dehn surgery space.
A non-abelian Seiberg-Witten invariant for integral homology 3-spheres.
A partial order in the knot table.
A rational noncommutative invariant of boundary links.