In 2018, Funakoshi, Hashizume, Ito, Kobayashi, and Murai used a deformation of spherical curves called deformation type $\alpha$. 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.

中文翻译：

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球面曲线和 Östlund 猜想的新变形

2018 年，Funakoshi、Hashizume、Ito、Kobayashi 和 Murai 使用了一种称为变形类型 $\alpha$ 的球面曲线变形。然后证明，如果两条球面曲线$P$和$P'$在RI型和RIII型变形至环境同位素组成的关系下是等价的，并且满足一定条件，则$P'$由$P$ 由 $\alpha$ 类型的有限变形序列组成。在本文中，我们介绍了一种新型的球面曲线变形，称为 $\beta$ 类型的变形。本文的主要结果是：当且仅当减少（P）和减少的（P'）通过恰好一个 RIII 类型的变形相互转换，