New deformations on spherical curves and Ostlund conjecture

In [1], a deformation of spherical curves called deformation type alpha was introduced. Then, it was showed that if two spherical curves P and P' are equivalent under the relation consisting of deformations of type RI and type RIII up to ambient isotopy, and satisfy certain conditions, then P' is obtained from P by a finite sequence of deformations of type alpha. In this paper, we introduce a new type of deformations of spherical curves, called deformation of type beta. The main result of this paper is: Two spherical curves P and P' are equivalent under (possibly empty) deformations of type RI and a single deformation of type RIII up to ambient isotopy if and only if reduced(P) and reduced(P') are transformed each other by exactly one deformation which is of type RIII, type alpha, or type beta up to ambient isotopy, where reduced(Q) is the spherical curve which does not contain a 1-gon obtained from a spherical curve Q by applying deformations of type RI up to ambient isotopy. (C) 2020 Published by Elsevier B.V.