Khovanov-Rozansky cycle calculus for bipartite links

TL;DR

This paper introduces a simplified bipartite calculus for a specific class of knots, generalizing Khovanov calculus to arbitrary N without complex matrix factorizations, facilitating the study of superpolynomials.

Contribution

It develops a bipartite calculus that simplifies Khovanov-Rozansky computations for antiparallel lock tangles, avoiding complex matrix factorizations.

Findings

Reproduces Khovanov-Rozansky polynomials in known cases

Simplifies the calculation process for bipartite links

Provides an accessible tool for superpolynomial studies

Abstract

Bipartite calculus is a direct generalization of Kauffman planar expansion from $N=2$ to arbitrary $N$, applicable to the restricted class of knots which are entirely made of antiparallel lock tangles. Whenever applicable, it allows a straightforward generalization of the Khovanov calculus without a need of the technically complicated matrix factorization used for arbitrary $N$ in the Khovanov-Rozansky (KR) approach. The main object here is the $3^{n}$-dimensional hypercube with $n$ being the number of bipartite vertices. Maps, differentials, complex and Poincar\'e polynomials are straightforward and indeed reproduce the Khovanov-Rozansky polynomials in the known cases. This provides a great simplification of the Khovanov-Rozansky calculus on the bipartite locus, what can make it an accessible tool for the study of superpolynomials.