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Skew characteristic polynomial of graphs and embedded graphs

R. Dogra, S. Lando

Abstract

We introduce a new one-variable polynomial invariant of graphs, which we call the skew characteristic polynomial. For an oriented simple graph, this is just the characteristic polynomial of its anti-symmetric adjacency matrix. For nonoriented simple graphs the definition is different, but for a certain class of graphs (namely, for intersection graphs of chord diagrams), it gives the same answer if we endow such a graph with an orientation induced by the chord diagram. We prove that this invariant satisfies Vassiliev's $4$-term relations and determines therefore a finite type knot invariant. We investigate the behaviour of the polynomial with respect to the Hopf algebra structure on the space of graphs and show that it takes a constant value on any primitive element in this Hopf algebra. We also provide a two-variable extension of the skew characteristic polynomial to embedded graphs and delta-matroids. The $4$-term relations for the extended polynomial prove that it determines a finite type invariant of multicomponent links.

Skew characteristic polynomial of graphs and embedded graphs

Abstract

We introduce a new one-variable polynomial invariant of graphs, which we call the skew characteristic polynomial. For an oriented simple graph, this is just the characteristic polynomial of its anti-symmetric adjacency matrix. For nonoriented simple graphs the definition is different, but for a certain class of graphs (namely, for intersection graphs of chord diagrams), it gives the same answer if we endow such a graph with an orientation induced by the chord diagram. We prove that this invariant satisfies Vassiliev's -term relations and determines therefore a finite type knot invariant. We investigate the behaviour of the polynomial with respect to the Hopf algebra structure on the space of graphs and show that it takes a constant value on any primitive element in this Hopf algebra. We also provide a two-variable extension of the skew characteristic polynomial to embedded graphs and delta-matroids. The -term relations for the extended polynomial prove that it determines a finite type invariant of multicomponent links.
Paper Structure (18 sections, 56 equations, 3 figures)

This paper contains 18 sections, 56 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Skew characteristic polynomial for graphs
  3. Definition of the skew characteristic polynomial
  4. Hopf algebra of graphs
  5. Justification: skew characteristic polynomial for intersection graphs of chord diagrams
  6. Examples
  7. Refined skew characteristic polynomial
  8. 4-term relations and knot invariants
  9. 4-term relations for chord diagrams and matrices
  10. 4-term relation for graphs
  11. Characteristic polynomial of antisymmetric matrices and 4-term relations
  12. Skew characteristic polynomial for delta-matroids and embedded graphs
  13. Embedded graphs
  14. Delta-matroids of graphs and embedded graphs
  15. The Hopf algebra of delta-matroids
  16. ...and 3 more sections

Figures (3)

  • Figure 1: The $5$-wheel and the $3$-prism
  • Figure 2: $4$-term relations for the $5$-wheel expressed in terms of intersection graphs; the value of the skew characteristic polynomial is indicated
  • Figure 3: $4$-term relations for the $5$-wheel expressing them in terms of intersection graphs; the value of the skew characteristic polynomial is indicated

Theorems & Definitions (5)

  • proof
  • proof
  • proof
  • proof
  • proof