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Polynomial graph invariants induced from the ${\mathfrak gl}$-weight system

N. Kodaneva, S. Lando

Abstract

Weight systems are functions on chord diagrams satisfying so-called Vassiliev's $4$-term relations. They are closely related to finite type knot invariants introduced by Vassiliev. Certain weight systems can be derived from graph invariants. Another main source of weight systems are Lie algebras. In recent papers, the weight systems associated to Lie algebras ${\mathfrak gl}(N)$ were unified in a universal ${\mathfrak gl}$-weight system, which takes values in the ring of polynomials in infinitely many variables. The unification has been achieved by extending the ${\mathfrak gl}(N)$-weight systems from chord diagrams to arbitrary permutations. A natural question then arises, namely, which already known weight systems can be obtained from the universal ${\mathfrak gl}$ weight system. In addition to understanding the internal relationship between weight systems, knowing that a given weight system can be induced from the ${\mathfrak gl}$-weight system would immediately lead to extending the former to arbitrary permutations. To each chord diagram, one can associate a graph, called the intersection graph of the chord diagram. Certain weight systems are completely determined by the intersection graphs. In general, the relationship between Lie algebra weight systems and polynomial graph invariants looks rather complicated. Our main result in the present paper consists in showing that the well-known graph and delta-matroid invariant, the interlace polynomial, can be induced from the universal ${\mathfrak gl}$-weight system. We provide an explicit substitution making the ${\mathfrak gl}$-weight system into the interlace polynomial for chord diagrams and their intersection graphs.

Polynomial graph invariants induced from the ${\mathfrak gl}$-weight system

Abstract

Weight systems are functions on chord diagrams satisfying so-called Vassiliev's -term relations. They are closely related to finite type knot invariants introduced by Vassiliev. Certain weight systems can be derived from graph invariants. Another main source of weight systems are Lie algebras. In recent papers, the weight systems associated to Lie algebras were unified in a universal -weight system, which takes values in the ring of polynomials in infinitely many variables. The unification has been achieved by extending the -weight systems from chord diagrams to arbitrary permutations. A natural question then arises, namely, which already known weight systems can be obtained from the universal weight system. In addition to understanding the internal relationship between weight systems, knowing that a given weight system can be induced from the -weight system would immediately lead to extending the former to arbitrary permutations. To each chord diagram, one can associate a graph, called the intersection graph of the chord diagram. Certain weight systems are completely determined by the intersection graphs. In general, the relationship between Lie algebra weight systems and polynomial graph invariants looks rather complicated. Our main result in the present paper consists in showing that the well-known graph and delta-matroid invariant, the interlace polynomial, can be induced from the universal -weight system. We provide an explicit substitution making the -weight system into the interlace polynomial for chord diagrams and their intersection graphs.
Paper Structure (17 sections, 19 theorems, 44 equations, 2 figures)

This paper contains 17 sections, 19 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $4$-term relations and weight systems
  3. Graphs, embedded graphs, and their delta-matroids
  4. ${\mathfrak{gl}}$-weight system
  5. Refined skew-characteristic polynomial
  6. Interlace polynomial
  7. A two-variable family induced from the ${\mathfrak{gl}}$-weight system
  8. Definition of the family
  9. Properties of $F_\epsilon$
  10. Proof of the main results
  11. Proof of Theorem \ref{['t-fe']}
  12. Proof of Theorem \ref{['t-rsc']}
  13. Proof of Theorem \ref{['t-is']}
  14. Odds and ends
  15. Hopf algebra structure on the spaces of graphs, chord diagrams, and permutations
  16. ...and 2 more sections

Key Result

Lemma 1.2

Let $\Gamma$ be an embedded graph, and let $D(\Gamma)=(E(\Gamma);\Phi(\Gamma))$ be its delta-matroid. Then, for a subset $E'\subset E(\Gamma)$, we have where $f(\Gamma_{E'})$ is the number of faces of the spanning embedded subgraph $\Gamma|_{E'}$.

Figures (2)

  • Figure 1: $4$-term relation
  • Figure 2: A $4$-term relation for chord diagrams and the corresponding intersection graphs

Theorems & Definitions (28)

  • Remark 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Theorem 1.4: Zhuoke Yang, ZY22
  • Example 1.5
  • Corollary 1.6
  • Example 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Example 1.10
  • ...and 18 more