  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.
