Markov theorem for doodles on two-sphere

TL;DR

This paper extends classical knot theory results to doodles on a two-sphere, establishing a Markov-type theorem that characterizes when two twins have equivalent closures.

Contribution

It provides a description of twins with equivalent closures, serving as an analogue of the classical Markov theorem for doodles on the two-sphere.

Findings

Established a Markov theorem analogue for doodles on the two-sphere.

Characterized when two twins have equivalent closures.

Extended classical knot theory results to doodles on spheres.

Abstract

In 1997 M.~Khovanov proved that any doodle can be presented as closure of twin, this result is analogue of classical Alexander's theorem for braids and links. We give a description of twins that have equivalent closures, this theorem is analogue of classical Markov theorem.