Khovanov-Rozansky cycle calculus for bipartite links
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915608082251776 |
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| author | Anokhina, A. Lanina, E. Morozov, A. |
| author_facet | Anokhina, A. Lanina, E. Morozov, A. |
| contents | 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 Poincaré 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_08721 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Khovanov-Rozansky cycle calculus for bipartite links Anokhina, A. Lanina, E. Morozov, A. High Energy Physics - Theory Mathematical Physics Geometric Topology 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 Poincaré 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. |
| title | Khovanov-Rozansky cycle calculus for bipartite links |
| topic | High Energy Physics - Theory Mathematical Physics Geometric Topology |
| url | https://arxiv.org/abs/2506.08721 |