The Expansion of a Chord Diagram and the Tutte Polynomial

Abstract

A chord diagram is a set of chords of a circle in which no pair of chords has a common endvertex. For a chord diagram E having a crossing S={x1x3,x2x4}, the expansion of E with respect to S is to replace E with one of two chord diagrams E1=(E\\S)∪{x2x3,x4x1} or E2=(E\\S)∪{x1x2,x3x4}. For a chord diagram E, the chord expansion number f(E) of E is defined as the cardinality of the multiset of all nonintersecting chord diagrams generated from E with a finite sequence of expansions. In this paper, it is shown that the chord expansion number is computed by the Tutte polynomial at the point (2,−1).