Is every knot isotopic to the unknot?

TL;DR

This paper proves that the Bing sling knot is not isotopic to any piecewise-linear (PL) knot, providing new insights into knot isotopy and the structure of knots in three-dimensional space.

Contribution

The paper demonstrates that the Bing sling cannot be isotopic to a PL knot, using novel isotopy constructions and algebraic conditions involving fundamental groups and handlebodies.

Findings

Bing sling is not isotopic to any PL knot.

Isotopy extends to 2-component links with linking number 1.

Results hold for boundary-link-like handlebodies with specific fundamental group properties.

Abstract

In 1974, D. Rolfsen asked: Is every knot in $S^3$ isotopic (=homotopic through embeddings) to a PL knot or, equivalently, to the unknot? In particular, is the Bing sling isotopic to a PL knot? We show that the Bing sling is not isotopic to any PL knot: (1) by an isotopy which extends to an isotopy of $2$-component links with $lk=1$; (2) through knots that are intersections of nested sequences of solid tori. There are also stronger versions of these results. In (1), the additional component may be allowed to self-intersect, and even to get replaced by a new one as long as it represents the same conjugacy class in $G/[G',G'']$, where $G$ is the fundamental group of the complement to the original component. In (2), the "solid tori" can be replaced by "boundary-link-like handlebodies", where a handlebody $V\subset S^3$ of genus $g$ is called boundary-link-like if $\pi_1(\overline{S^3-V})$ admits a homomorphism to the free group $F_g$ such that the composition $\pi_1(\partial V)\to\pi_1(\overline{S^3-V})\to F_g$ is surjective.