Skew characteristic polynomial of graphs and embedded graphs

Riya Dogra, Sergei Lando

Research output: Contribution to JournalArticleAcademicpeer-review

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 non-oriented 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 behavior 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 multi-component links.

Original languageEnglish
Pages (from-to)87-111
Number of pages25
JournalCommunications in Mathematics
Volume31
Issue number3
Early online date30 Dec 2023
DOIs
Publication statusPublished - 2023

Bibliographical note

Special issue: in memory of Sergei Duzhin.

Publisher Copyright:
© 2023 Riya Dogra and Sergei Lando.

Keywords

  • 4-term relations
  • Characteristic polynomial
  • Delta-matroid
  • Graph polynomials
  • Knot invariants
  • Weight system

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