||Journal of Gökova Geometry Topology, Volume 1 (2007)|
||Virtual crossings, convolutions and a categorification of the SO(2N) Kauffman polynomial|
||Mikhail Khovanov and Lev Rozansky|
We suggest a categorification procedure for the SO(2N) one-variable
specialization of the two-variable Kauffman polynomial.
The construction has many similarities with the HOMFLY-PT
categorification: a planar graph formula for the polynomial is
converted into a complex of graded vector spaces, each of them being
homology of a Z2-graded differential vector space associated to a graph
and constructed using matrix factorizations.
This time, however, the elementary matrix factorizations are not Koszul; instead, they are
convolutions of of chain complexes of Koszul matrix factorizations.
We prove that the homotopy class of the resulting complex
associated to a diagram of a link is invariant under the first two
Reidemeister moves and conjecture its invariance
under the third move.
|Submitted: ||January 23, 2007|
|Revised: ||November 23, 2007|
|Accepted: ||November 24, 2007|
Journal of GGT Volume 1|
Last updated: December 2007
Wed address: GokovaGT.org/journal/2007