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

Sergei Lando, Zhuoke Yang

Abstract

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.

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

Abstract

In a recent paper by M.Kazarian and the second author, a recurrence for the Lie algebras weight systems has been suggested; the recurrence allows one to construct the universal weight system. The construction is based on an extension of the weight systems to permutations. Another recent paper, by M. Kazarian, N. Kodaneva, and the first author, shows that under the substitution for the Casimir elements , the leading term in of the value of the universal 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 weight system. That is, we show that the leading term of the universal weight system also becomes the chromatic polynomial under a specific substitution.

Paper Structure

This paper contains 10 sections, 10 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Construction of the universal ${\mathfrak{so}}$ weight system
  3. Definition of $w_{\mathfrak{so}}$
  4. Lie algebra ${\mathfrak{so}}(N)$ with standard representation
  5. Recurrence rule for pre-chromatic substitution
  6. Hopf algebras of permutations
  7. Prechromatic substitution
  8. Values of $Y_0$ on cyclic permutations
  9. Chromatic substitution
  10. Proof of Theorem \ref{['th-csp']}

Key Result

Theorem 2.3

BN95 For a chord diagram $D$ of order $n$, where the sum is taken over all $2^n$ states for $D$.

Theorems & Definitions (19)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • Definition 3.1
  • Definition 3.2: prechromatic substitution
  • Lemma 3.3
  • Theorem 3.4: Recurrence rule for pre-chromatic substitution
  • ...and 9 more