Calculating Bar-Natan's Characteristic Two Khovanov Homology

Abstract

ABSTRACT. 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 sequences. The E 2 -pages can be described in terms of groups arising from the action of a certain endomorphism on F 2 -Khovanov homology. Some simple consequences are discussed.

A spectral sequence for Khovanov homology with an application to (3,q)–torus links

“…By indexing as for Khovanov homology the differential d i on the E i -page is of bidegree .1; 2/. Moreover d 1 can be understood explicitly as explained in [8]. Thus, we can again attempt to bypass explicit computation of differentials in the (reduced version of the) spectral sequence of Proposition 2.2 by the same method as above, but now using reduced Bar-Natan theory.…”

The universalsl3–link homology

“…In [8], following his own seminal work in [6] and Lee [11], Bar-Natan [2] and Turner's [13] subsequent contributions, Khovanov classified all possible Frobenius systems of dimension two which give rise to link homologies via his construction in [6] and showed that there is a universal one, given by ‫ޚ‬OEX; a; b=.X Working over ‫,ރ‬ one can take a and b to be complex numbers and study the corresponding homology with coefficients in ‫.ރ‬ We refer to the latter as the sl 2 -link homologies over ‫,ރ‬ because they are all deformations of Khovanov's original link homology whose Euler characteristic equals the Jones polynomial which is well known to be related to the Lie algebra sl 2 [8]. Using the ideas in [8; 11; 13], the authors and Turner showed in [12] that there are only two isomorphism classes of sl 2 -link homologies over ‫.ރ‬ Given a; b 2 ‫,ރ‬ the isomorphism class of the corresponding link homology is completely determined by the number of distinct roots of the polynomial X 2 aX b .…”

On transverse invariants from Khovanov homology

“…⊗|K| (where |K| denotes the number of components of K) [Tur06]. Moreover, the generators of H * (C(K)) correspond to orientations of K, as follows.…”