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

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.

References in corpus (3)

• Some differentials on Khovanov-Rozansky homology
• Matrix Factorizations and Kauffman Homology
• Braids, Transversal links and the Khovanov-Rozansky Theory

Cited by in corpus (22)

• Link Homologies and the Refined Topological Vertex
• Colored HOMFLY polynomials for the pretzel knots and links
• Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
• Colored Kauffman Homology and Super-A-polynomials
• Introduction to Khovanov Homologies. I. Unreduced Jones superpolynomial
• Orientifolds and the Refined Topological String
• HOMFLY polynomials in representation [3,1] for 3-strand braids
• Generic deformations of the colored sl(N)-homology for links
• Checks of integrality properties in topological strings
• Matrix factorizations and intertwiners of the fundamental representations of quantum group U_q (sl_n)
• Matrix factorizations and colored MOY graphs
• Extending Landau-Ginzburg models to the point
• An Introduction to Khovanov Homology
• On the Kauffman-Vogel and the Murakami-Ohtsuki-Yamada Graph Polynomials
• Colored sl(N) link homology via matrix factorizations
• Matrix factorizations and double line in ${\mathfrak{sl}}_n$ quantum link invariant
• HOMFLY-PT homology for general link diagrams and braidlike isotopy
• A categorification of the quantum sl(N)-link polynomials using foams
• Khovanov-Rozansky Homology and Conway Mutation
• Virtual crossings and a filtration of the triply graded homology of a link diagram
• A skein approach to Bennequin type inequalities
• A remark on BMW algebra, q-Schur algebras and categorification