Unknotting submanifolds of the 3-sphere by twistings
Abstract
By the Fox's re-embedding theorem, any compact submanifold of the 3-sphere can be re-embedded in the 3-sphere so that it is unknotted. It is unknown whether the Fox's re-embedding can be replaced with twistings. In this paper, we will show that any closed 2-manifold embedded in the 3-sphere can be unknotted by twistings. In spite of this phenomenon, we show that there exists a compact 3-submanifold of the 3-sphere which cannot be unknotted by twistings. This shows that the Fox's re-embedding cannot always be replaced with twistings.
Cite
@article{arxiv.1609.06573,
title = {Unknotting submanifolds of the 3-sphere by twistings},
author = {Makoto Ozawa},
journal= {arXiv preprint arXiv:1609.06573},
year = {2019}
}
Comments
8 pages, 5 figures. In v2, more details on the Proof of Theorem 2.6 are given, and concluding remarks are added. In v3, Definitions 1.1 and 1.2 are clarified, and Corollary 2.4 is deleted by the ambiguous statement