Transition polynomial as a weight system for binary delta-matroids

To a singular knot $K$ with $n$ double points, one can associate a chord diagram with $n$ chords. A chord diagram can also be understood as a $4$-regular graph endowed with an oriented Euler circuit. L. Traldi introduced a polynomial invariant for such graphs, called a transition polynomial. We specialize this polynomial to a multiplicative weight system, that is, a function on chord diagrams satisfying $4$-term relations and determining thus a finite type knot invariant. We prove a similar statement for the transition polynomial of general ribbon graphs and binary delta-matroids defined by R. Brijder and H. J. Hoogeboom, which defines, as a consequence, a finite type invariant of links.