Space of chord diagrams on spherical curves

Abstract

In this paper, we give a definition of $\mathbb{Z}$-valued functions from the ambient isotopy classes of spherical/plane curves derived from chord diagrams, denoted by $\sum_i \alpha_i x_i$. Then, we introduce certain elements of the free $\mathbb{Z}$-module generated by the chord diagrams with at most $l$ chords, called relators of Type (I) ((SII), (WII), (SIII), or (WIII), resp.), and introduce another function $\sum_i \alpha_i \tilde{x}_i$ derived from $\sum_i \alpha_i x_i$. The main result (Theorem~1) shows that if $\sum_i \alpha_i \tilde{x}_i$ vanishes for the relators of Type (I) ((SII), (WII), (SIII), or (WIII), resp.), then $\sum_i \alpha_i x_i$ is invariant under the Reidemeister move of type RI (strong RII, weak RII, strong RIII, or weak RIII, resp.) that is defined in [Ito-Takimura (2013), J. Knot Theory Ramifications].