Arrow diagrams on spherical curves and computations

Abstract

We give a definition of an integer-valued function [Formula: see text] derived from arrow diagrams for the ambient isotopy classes of oriented spherical curves. Then, we introduce certain elements of the free [Formula: see text]-module generated by the arrow diagrams with at most [Formula: see text] arrows, called relators of Type ([Formula: see text]) (([Formula: see text]), ([Formula: see text]), ([Formula: see text]) or ([Formula: see text]), respectively), and introduce another function [Formula: see text] to obtain [Formula: see text]. One of the main results shows that if [Formula: see text] vanishes on finitely many relators of Type ([Formula: see text]) (([Formula: see text]), ([Formula: see text]), ([Formula: see text]) or ([Formula: see text]), respectively), then [Formula: see text] is invariant under the deformation of type RI (strong RI[Formula: see text]I, weak RI[Formula: see text]I, strong RI[Formula: see text]I[Formula: see text]I or weak RI[Formula: see text]I[Formula: see text]I, respectively). The other main result is that we obtain new functions of arrow diagrams with up to six arrows explicitly. This computation is done with the aid of computers.

Goussarov-Polyak-Viro Conjecture for degree three case

“…We mimic ideas of [5], but the definition of arrow diagrams is slightly different from that of [5] (cf. [4,7,9]). Definition 2 (Reidemeister moves on arrow diagrams, cf.…”