Uniform Convergence and Knot Equivalence

Given a uniformly convergent sequence of ambient isotopies $(H_n{)}_{n\in \mathbb{N}}$, bijectivity of the limit function $H_{\mathrm{\infty }}$ together with a minor compactness condition guarantees that $H_{\mathrm{\infty }}$ 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.