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The universal gl-weight system and the chromatic polynomial
Journal of Geometry and Physics. 2026. No. 225. Article 105841.
Weight systems associated to the Lie algebras 𝔤𝔩(N) for N = 1,2,... can be unified into auniversal one. The construction is based on an extension of the 𝔤𝔩(N) weight systems to permutations. This universal weight system takes values in the algebra of polynomials C[N;C1,C2,...] in infinitely many variables. We show that under the substitution Cm = xNm−1, m = 1,2,..., the leading term in N of the value of the universal 𝔤𝔩 weight system becomes the chromatic polynomial of the intersection graph of the chord diagram. Moreover, we show that under the substitution Cm = pmNm−1, m = 1,2,..., the leading term in N of the value of the universal 𝔤𝔩-weight system determines a filtered Hopf algebra homomorphism from the rotational Hopf algebra of permutations, which we construct in the present paper, to the Hopf algebra of polynomials C[p1,p2,...].
Keywords: Hopf algebraweight systemchord diagramLie algebrachromatic polynomialFinite type knot invariants
Publication based on the results of:
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