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arXiv · 1809.03431

An extension of Stanley's chromatic symmetric function to binary delta-matroids

Marina Dudina·Vyacheslav Zhukov
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Abstract

Stanley's symmetrized chromatic polynomial is a generalization of the ordinary chromatic polynomial to a graph invariant with values in a ring of polynomials in infnitely many variables. The ordinary chromatic polynomial is a specialization of the Stanley's one. Our goal is to extend Stanley's chromatic polynomial to embedded graphs. In contrast to well-known extensions of, say, the Tutte polynomial from abstract to embedded graphs [5], we do not treat an embedded graph as an abstract graph endowed with additional information about the embedding. Instead, we consider the binary delta-matroid associated to an embedded graph and define the extended Stanley's chromatic polynomial as an invariant of delta-matroids. We show that, similarly to Stanley's symmetrized chromatic polynomial of graphs, which satisfies 4-term relations for simple graphs and determines in this way a knot invariant, extended Stanley's chromatic polynomial of binary delta-matroids we define satisfies the 4-term relations for binary delta-matroids [10] and determines, therefore, an invariant of links.

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Marina Dudina, Vyacheslav Zhukov. 2018-09-10. An extension of Stanley's chromatic symmetric function to binary delta-matroids. https://arxiv.org/abs/1809.03431

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