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arXiv · 2506.08721

Khovanov-Rozansky cycle calculus for bipartite links

A. Anokhina·E. Lanina·A. Morozov
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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.

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A. Anokhina, E. Lanina, A. Morozov. 2025-06-10. Khovanov-Rozansky cycle calculus for bipartite links. https://doi.org/10.1140/epjc%2Fs10052-025-14854-y

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