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. Contents 1. Introduction 2. Background 2.1.