Uniform Convergence and Knot Equivalence
Abstract
Given a uniformly convergent sequence of ambient isotopies (Hn)n∈N, bijectivity of the limit function H∞ together with a minor compactness condition guarantees that H∞ 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.
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.