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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2411.01128 |
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| _version_ | 1866917827315761152 |
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| author | Lando, Sergei Yang, Zhuoke |
| author_facet | Lando, Sergei Yang, Zhuoke |
| contents | In a recent paper by M.Kazarian and the second author, a recurrence for the Lie algebras $\mathfrak{so}(N)$ weight systems has been suggested; the recurrence allows one to construct the universal $\mathfrak{so}$ weight system. The construction is based on an extension of the $\mathfrak{so}$ weight systems to permutations. Another recent paper, by M. Kazarian, N. Kodaneva, and the first author, shows that under the substitution $C_m=xN^{m-1}, m=1,2,\dots,$ for the Casimir elements $C_m$, the leading term in $N$ of the value of the universal $\mathfrak{gl}$ weight system becomes the chromatic polynomial of the intersection graph of the chord diagram. In the present paper, we establish a similar result for the universal $\mathfrak{so}$ weight system. That is, we show that the leading term of the universal $\mathfrak{so}$ weight system also becomes the chromatic polynomial under a specific substitution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_01128 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Chromatic polynomial and the $\mathfrak{so}$ weight system Lando, Sergei Yang, Zhuoke Combinatorics In a recent paper by M.Kazarian and the second author, a recurrence for the Lie algebras $\mathfrak{so}(N)$ weight systems has been suggested; the recurrence allows one to construct the universal $\mathfrak{so}$ weight system. The construction is based on an extension of the $\mathfrak{so}$ weight systems to permutations. Another recent paper, by M. Kazarian, N. Kodaneva, and the first author, shows that under the substitution $C_m=xN^{m-1}, m=1,2,\dots,$ for the Casimir elements $C_m$, the leading term in $N$ of the value of the universal $\mathfrak{gl}$ weight system becomes the chromatic polynomial of the intersection graph of the chord diagram. In the present paper, we establish a similar result for the universal $\mathfrak{so}$ weight system. That is, we show that the leading term of the universal $\mathfrak{so}$ weight system also becomes the chromatic polynomial under a specific substitution. |
| title | Chromatic polynomial and the $\mathfrak{so}$ weight system |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.01128 |