# Moscow Mathematical Journal

Volume 22, Issue 1, January–March 2022 pp. 69–81.

Transition Polynomial as a Weight System for Binary Delta-Matroids

**Authors**:
Alexander Dunaykin (1) and Vyacheslav Zhukov (2)

**Author institution:**(1) International Laboratory of Cluster Geometry National Research University Higher School of Economics

(2) International Laboratory of Cluster Geometry National Research University\linebreak Higher School of Economics

**Summary: **

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.

2020 Math. Subj. Class. 05C31

**Keywords:**Knot, link, finite type invariant of knots, chord diagram, transition polynomial, delta-matroid.

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