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

Mikhail Khovanov·Lev Rozansky
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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.

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Mikhail Khovanov, Lev Rozansky. 2007-01-12. Virtual crossings, convolutions and a categorification of the SO(2N) Kauffman polynomial. https://arxiv.org/abs/math/0701333

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