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 language | English |
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Pages (from-to) | 87-111 |
Number of pages | 25 |
Journal | Communications in Mathematics |
Volume | 31 |
Issue number | 3 |
Early online date | 30 Dec 2023 |
DOIs | |
Publication status | Published - 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