Khovanov--Rozansky matrix factorization reduction for bipartite links

Abstract

The Khovanov-Rozansky (KR) link polynomial is a certain $t$-deformation of Wilson loops in 3-dimensional $SU(N)$ Chern--Simons topological field theory, believed to be an observable in the refined Chern-Simons theory, probably described in terms of 4d or 5d QFT and related by a certain procedure to the triply-graded link superpolynomial. This link invariant was originally introduced by M. Khovanov and L. Rozansky through a sophisticated matrix factorization technique based on the bicomplex structure, which depends on entire link diagrams and rapidly increases in complexity with the growth of a link. However, for particular link diagrams a local reduction is possible, allowing to eliminate vertices in a regular way, and thus, simplifying the KR polynomial and making it as simple as the Khovanov polynomial in the $N=2$ case. In particular, for a distinguished family of bipartite links, matrix factorization defined on MOY diagrams reduces just to planar cycles - very similar to the original Kauffman-Khovanov construction at $N=2$ for the Jones polynomial and its $t$-deformation. In the bipartite case, this can be done for any $N$. We make a further step of simplification and reduce from cohomology factor-rings in even variables crucially depending on a MOY diagram to vector spaces spanned by odd variables, so that the initial bicomplex of matrix factorizations becomes a monocomplex of just tensor products of $N$-dimensional vector spaces. We also find the explicit form of three universal morphisms which were guessed in a recent paper on this subject. Universality means independence of the other edges of the diagram, and we explain why this works in this particular case.