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

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