Uniform Convergence and Knot Equivalence

TL;DR

The paper explores conditions under which uniform convergence of ambient isotopies ensures the limit is also an isotopy, and applies this to analyze countably-crossed tame knots versus wild curves.

Contribution

It introduces diagrammatic conditions replacing uniform convergence to guarantee ambient isotopy limits and constructs examples of tame knots with countably many crossings.

Findings

Uniform convergence plus compactness implies limit isotopy.

Diagrammatic conditions suffice for countably-many Reidemeister moves.

Examples differentiate tame knots with countably many crossings from wild curves.

Abstract

Given a uniformly convergent sequence of ambient isotopies $(H_n)_{n\in\mathbb{N}}$, bijectivity of the limit function $H_\infty$ together with a minor compactness condition guarantees that $H_\infty$ is also an ambient isotopy. By offloading the uniform convergence hypothesis to a more diagrammatic condition, we obtain sufficient conditions for performing countably-many Reidemeister moves. We use this to construct examples of tame knots with countably-many crossings and discuss what distinguishes these from similar-looking wild curves.