The universal gl-weight system and the chromatic polynomial

In a recent paper Zhuoke Yang, New approaches to gl(N) weight system, Izvestiya Mathematics, 2023, vol. 77:6, 150--166; arXiv:2202.12225 (2022) a construction of a weight system, which unifies gl(N) weight systems for N=1,2,, has been suggested. 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 substition Cₘ=pₘ Nᵐ⁻¹, m=1,2,, the leading term in N of the value of the universal gl weight system determines a flitered 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,].