Chromatic polynomial and the $\mathfrak{so}$ weight system

• Published in: Annales de l'Institut Henri Poincaré. D. Combinatorics, physics and their interactions
• Main Authors: Lando, Sergei, Yang, Zhuoke
• Format: Journal Article
• Language: English
• Published: 25.04.2025
• ISSN: 2308-5827; 2308-5835
• DOI: 10.4171/aihpd/208

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,\ldots, 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. The present paper establishes 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.