Reductions in Khovanov-Rozansky operator formalism

Abstract

Sophisticated Khovanov-Rozansky (KhR) description of knot invariants in the fundamental representation can be reformulated in terms of bicomplex with a simple physical meaning. Namely, the counterintuitive matrix factorization is substituted by simple operators $D$, locally constructed for every MOY resolution of a link diagram, which becomes nilpotent when the diagram has no external lines. Operators for different resolutions are related by equally simple conjugations $χ^{(\pm)}$. The KhR procedure then splits in two steps - defining ``vertical'' cohomologies of $D$, which are associated with particular resolutions and will be put at vertices of the hypercube, and conjugations $χ^{(\pm)}$, that define morphisms along its edges. As usual, standard combinations of morphisms are nilpotent, and one can define ``horizontal'' cohomologies - which are then combined into Poincaré polynomial, called KhR polynomial in application to links. This construction remains global in the sense that resulting cohomologies depend on the entire link diagram, but all its building blocks, including the operators and morphisms are local in the sense that they are defined for its particular vertices. Sometimes, this allows simple local reductions, allowing to eliminate or change particular vertices or sets of those. Along with the obvious case of Reidemeister equivalencies this happens also for antiparallel-lock tangles, what is responsible for simplification of bipartite calculus. In the $N=2$ and arbitrary $N$ bipartite cases, one can also provide global reductions transferring the local construction of the KhR double-complex to the global construction of the Khovanov(-like) single-complex.