Transition polynomial as a weight system for binary delta-matroids

2019/07/08 by Dunaykin, Alexander, Zhukov, Vyacheslav
#05C31 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1907.03831

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. 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.

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