Analogue of Goeritz matrices for computation of bipartite HOMFLY-PT polynomials
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909005417283584 |
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| author | Anokhina, A. Korzun, D. Lanina, E. Morozov, A. |
| author_facet | Anokhina, A. Korzun, D. Lanina, E. Morozov, A. |
| contents | The Goeritz matrix is an alternative to the Kauffman bracket and, in addition, makes it possible to calculate the Jones polynomial faster with some minimal choice of a checkerboard surface of a link diagram. We introduce a modification of the Goeritz method that generalizes the Goeritz matrix for computing the HOMFLY-PT polynomials for any $N$ in the special case of bipartite links. Our method reduces to purely algebraic operations on matrices, and therefore, it can be easily implemented as a computer program. Bipartite links form a rather large family including a special class of Montesinos links constructed from the so-called rational tangles. We demonstrate how to obtain a bipartite diagram of such links and calculate the corresponding HOMFLY-PT polynomials using our developed generalized Goeritz method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03116 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analogue of Goeritz matrices for computation of bipartite HOMFLY-PT polynomials Anokhina, A. Korzun, D. Lanina, E. Morozov, A. Geometric Topology High Energy Physics - Theory Mathematical Physics The Goeritz matrix is an alternative to the Kauffman bracket and, in addition, makes it possible to calculate the Jones polynomial faster with some minimal choice of a checkerboard surface of a link diagram. We introduce a modification of the Goeritz method that generalizes the Goeritz matrix for computing the HOMFLY-PT polynomials for any $N$ in the special case of bipartite links. Our method reduces to purely algebraic operations on matrices, and therefore, it can be easily implemented as a computer program. Bipartite links form a rather large family including a special class of Montesinos links constructed from the so-called rational tangles. We demonstrate how to obtain a bipartite diagram of such links and calculate the corresponding HOMFLY-PT polynomials using our developed generalized Goeritz method. |
| title | Analogue of Goeritz matrices for computation of bipartite HOMFLY-PT polynomials |
| topic | Geometric Topology High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2507.03116 |