New deformations on spherical curves and Östlund conjecture

In [1], a deformation of spherical curves called deformation type α 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 obtain...

Full description

Saved in:
Bibliographic Details
Published in:Topology and its applications 2021-09, Vol.301, p.107508, Article 107508
Main Authors: Hashizume, Megumi, Ito, Noboru
Format: Article
Language:English
Subjects:
Homotopy
Reidemeister move
Spherical curve
Östlund conjecture
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:In [1], a deformation of spherical curves called deformation type α 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 α. In this paper, we introduce a new type of deformations of spherical curves, called deformation of type β. 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 α, or type β 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.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2020.107508