Unknotting submanifolds of the 3-sphere by twistings

TL;DR

This paper investigates whether twistings can replace re-embedding in unknottings of submanifolds in the 3-sphere, showing that while some can be unknotted by twistings, others cannot, indicating limitations of twistings as a universal method.

Contribution

It demonstrates that twistings can unknot certain submanifolds but are insufficient for all, highlighting a fundamental limitation compared to re-embedding.

Findings

Closed 2-manifolds can be unknotted by twistings.

Existence of 3-submanifolds that cannot be unknotted 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.