R. Dogra ; S. Lando - Skew characteristic polynomial of graphs and embedded graphs

cm:11310 - Communications in Mathematics, December 30, 2023, Volume 31 (2023), Issue 3 (Special issue: in memory of Sergei Duzhin) - https://doi.org/10.46298/cm.11310
Skew characteristic polynomial of graphs and embedded graphsArticle

Authors: R. Dogra ; S. Lando

    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.


    Volume: Volume 31 (2023), Issue 3 (Special issue: in memory of Sergei Duzhin)
    Published on: December 30, 2023
    Accepted on: September 20, 2023
    Submitted on: May 14, 2023
    Keywords: Mathematics - Combinatorics

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