Transition Polynomial as a Weight System for Binary Delta-Matroids

Abstract

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. For a given 4-regular graph, we can build 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 knot invariant. We extend our function to ribbon graphs and further to binary delta-matroids and show that 4-term relations are satisfied for it.