The expansion of a chord diagram and the Tutte polynomial

Abstract

A chord diagram is a set of chords of a circle such that no pair of chords has a common endvertex. A chord diagram E is called nonintersecting if E contains no crossing. For a chord diagram E having a crossing S={x1x3,x2x4}, the expansion of E with respect to S is to replace E with E1=(E∖S)∪{x2x3,x4x1} or E2=(E∖S)∪{x1x2,x3x4}. For a chord diagram E, let f(E) be the chord expansion number of E, which 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 f(E) equals the value of the Tutte polynomial at the point (2,−1) for the interlace graph GE corresponding to E. The chord expansion number of a complete multipartite chord diagram is also studied. An extended abstract of the paper was published (Nakamigawa and Sakuma, 2017) [13].