# Arrow diagrams on spherical curves and computations

## TL;DR

This paper defines and investigates integer-valued functions derived from arrow diagrams on spherical curves, establishing invariance under certain deformations and computing functions with up to six arrows using computer assistance.

## Contribution

It introduces new invariants from arrow diagrams for spherical curves and proves their invariance under specific isotopies, including computational results for diagrams with up to six arrows.

## Key findings

- Invariants are established for spherical curves based on arrow diagrams.
- Invariance under specific deformation types is proven for these functions.
- Computational methods extend the analysis to diagrams with up to six arrows.

## Abstract

We give a definition of an integer-valued function $\sum_i \alpha_i x ^*_i$ derived from arrow diagrams for the ambient isotopy classes of oriented spherical curves. Then, we introduce certain elements of the free $\mathbb{Z}$-module generated by the arrow diagrams with at most $l $ arrows, called relators of Type~($\check{\rm{I}}$) (($\check{\rm{SI\!I} }$), ($\check{\rm{WI\!I}}$), ($\check{\rm{SI\!I\!I}}$), or ($\check{\rm{ WI\!I\!I}}$), resp.), and introduce another function $\sum_i \alpha_i \tilde{x}^*_i$ to obtain $\sum_i \alpha_i x^*_i$. One of the main results shows that if $\sum_i \alpha_i \tilde{x}^*_i$ vanishes on finitely many relators of Type~($\check{\rm{I}}$) (($\check{\rm{SI\!I}}$) , ($\check{\rm{WI\!I}}$), ($\check{\rm{SI\!I\!I}}$), or ($\check{\rm{WI\! I\!I}}$), resp.), then $\sum_i \alpha_i \tilde{x}$ is invariant under the deformation of type $\rm{RI}$ (strong$\rm{RI\!I}$, weak$\rm{RI\!I}$, strong$\rm{RI\!I\!I}$, or weak$\rm{RI\!I\!I}$, resp.). The other main result is that we obtain functions of arrow diagrams with up to six arrows. This computation is done with the aid of computers.