Khovanov--Rozansky matrix factorization reduction for bipartite links

TL;DR

This paper simplifies the computation of Khovanov-Rozansky link invariants for bipartite links by reducing complex matrix factorizations to more manageable forms, enabling explicit calculations for any N.

Contribution

It introduces a local reduction technique for KR matrix factorizations on bipartite links, simplifying them to tensor products of vector spaces and providing explicit universal morphisms.

Findings

Reduction of KR matrix factorizations to planar cycles for bipartite links

Simplification to vector spaces spanned by odd variables

Explicit form of three universal morphisms

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.