Skew characteristic polynomial of graphs and embedded graphs

Communications in Mathematics. 2023. Vol. 31. No. 3. P. 87–111.

Dogra R., Lando S.

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

Keywords: characteristic polynomial, knot invariants, delta-matroid, 4-term relations, graph polynomials, Weight system