تقرير
Skew characteristic polynomial of graphs and embedded graphs
| العنوان: | Skew characteristic polynomial of graphs and embedded graphs |
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| المؤلفون: | Dogra, R., Lando, S. |
| المصدر: | Communications in Mathematics, Volume 31 (2023), Issue 3 (Special issue: in memory of Sergei Duzhin) (December 30, 2023) cm:11310 |
| سنة النشر: | 2022 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Combinatorics |
| الوصف: | 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. |
| نوع الوثيقة: | Working Paper |
| DOI: | 10.46298/cm.11310 |
| URL الوصول: | http://arxiv.org/abs/2201.07084 |
| رقم الأكسشن: | edsarx.2201.07084 |
| قاعدة البيانات: | arXiv |
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