Knot diagrams on a punctured sphere as a model of string figures

TL;DR

This paper models string figures as knot diagrams on a punctured sphere, analyzing minimal crossings under Reidemeister moves that respect finger-like lines, providing a mathematical framework for understanding string figure complexity.

Contribution

It introduces a novel topological model of string figures using knot diagrams on a punctured sphere and studies minimal crossings under constrained Reidemeister moves.

Findings

Characterization of minimal crossings in the model

Insights into the topological complexity of string figures

Potential applications to understanding physical string manipulations

Abstract

A string figure is topologically a trivial knot lying on an imaginary plane orthogonal to the fingers with some crossings. The fingers prevent cancellation of these crossings. As a mathematical model of string figure we consider a knot diagram on the $xy$-plane in $xyz$-space missing some straight lines parallel to the $z$-axis. These straight lines correspond to fingers. We study minimal number of crossings of these knot diagrams under Reidemeister moves missing these lines.