Sub-chord diagrams of knot projections

Abstract

A chord diagram is a circle with paired points with each pair of points connected by a chord. Every generic immersed spherical curve provides a chord diagram by associating each chord with two preimages of a double point. Any two spherical curves can be related by a finite sequence of three types of local replacement RI, RII, and RIII, called Reidemeister moves. This study counts the difference in the numbers of sub-chord diagrams embedded in a full chord diagram of any spherical curve by applying one of the moves RI, strong RII, weak RII, strong RIII, and weak RIII defined by connections of branches related to the local replacements (Theorem 1). This yields a new integer-valued invariant under RI and strong RIII that provides a complete classification of prime reduced spherical curves with up to at least seven double points (Theorem 2, Fig. 24): there has been no such invariant before. The invariant expresses the necessary and sufficient condition that spherical curves can be related to a simple closed curve by a finite sequence of RI and strong RIII moves (Theorem 3). Moreover, invariants of spherical curves under flypes are provided by counting sub-chord diagrams (Theorem 4).