Space of chord diagrams on spherical curves

TL;DR

This paper introduces a new framework for understanding functions on spherical curves derived from chord diagrams, establishing conditions under which these functions are invariant under Reidemeister moves.

Contribution

It defines a class of $bZ$-valued functions on spherical curves and identifies conditions for their invariance under Reidemeister moves, extending previous knot theory invariants.

Findings

Defines $bZ$-valued functions from chord diagrams on spherical curves.

Introduces relators of various types to analyze invariance.

Proves that certain functions vanish on relators imply invariance under Reidemeister moves.

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].