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

Abstract: Weight systems are functions on chord diagrams satisfying the so-called Vassiliev 4-term relations. They are closely related to finite-type knot invariants. Certain weight systems can be derived from graph invariants and Lie algebra. In a recent paper by M. Kazarian and the speaker, 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 S. Lando, shows that under the specific substitution for the Casimir elements, the leading term in $N$ of the universal $\mathfrak{gl}$ weight system becomes the chromatic polynomial of the intersection graph of the chord diagram. In this talk, we establish a similar result for the universal $\mathfrak{so}$ weight system. that is the leading term of the universal $\mathfrak{so}$ weight system also becomes the chromatic polynomial under a specific substitution.

The talk is based on a joint work arxiv: 2411.01128 with Sergei Lando.

mathematical physicsdynamical systemsquantum algebrarepresentation theorysymplectic geometry

Audience: general audience

( slides | video )

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BIMSA Integrable Systems Seminar

Series comments: The aim is to bring together experts in integrable systems and related areas of theoretical and mathematical physics and mathematics. There will be research presentations and overview talks.

Audience: Graduate students and researchers interested in integrable systems and related mathematical structures, such as symplectic and Poisson geometry and representation theory.

The zoom link will be distributed by mail, so please join the mailing list if you are interested in attending the seminar.

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