Virtual crossings, convolutions and a categorification of the SO(2N) Kauffman polynomial

TL;DR

This paper introduces a new categorification of the SO(2N) specialization of the Kauffman polynomial, using graph-based complexes and matrix factorizations, extending methods from HOMFLYPT categorification.

Contribution

It develops a novel categorification framework employing convolutions of Koszul matrix factorizations, and proves invariance under Reidemeister moves I and II for the SO(2N) Kauffman polynomial.

Findings

Constructs a complex of graded vector spaces from planar graphs.

Proves invariance under Reidemeister moves I and II.

Conjectures invariance under Reidemeister move III.

Abstract

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 HOMFLYPT categorification: a planar graph formula for the polynomial is converted into a complex of graded vector spaces, each of them being the homology of a Z_2 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 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.