Topological and physical knot theory are distinct

TL;DR

This paper demonstrates that physical knot theory, which considers fixed length and thickness, differs from classical knot theory, providing examples like the Gordian Split Link that cannot be separated physically despite being split classically.

Contribution

It introduces the concept that physical isotopy classes can differ from classical ones and presents explicit examples illustrating this distinction.

Findings

Physical isotopy classes can differ from classical knot classes.

The Gordian Split Link cannot be physically separated despite being classically split.

Physical constraints alter the topological classification of knots and links.

Abstract

Physical knots and links are one-dimensional submanifolds of R^3 with fixed length and thickness. We show that isotopy classes in this category can differ from those of classical knot and link theory. In particular we exhibit a Gordian Split Link, a two component link that is split in the classical theory but cannot be split with a physical isotopy.