A constructive solution to the equivalence problem for knot projectivizations

TL;DR

This paper presents an explicit algorithmic method to determine if different projectivizations of the same affine knot are equivalent in real projective space, extending to lens spaces and providing concrete isotopies.

Contribution

It introduces a constructive, algorithmic solution to the knot projectivization equivalence problem, including a generalization to lens spaces and explicit isotopies.

Findings

Provided an explicit isotopy algorithm for affine knot projectivizations.

Extended the method to lens spaces, including real projective space.

Applied the algorithm to known knot pairs, resolving open questions.

Abstract

The problem of whether different projectivizations of the same affine knot $K\subset\mathbb{S}^3$ are equivalent in $\mathbb{R}\mathbb{P}^3$ can be found in [11] and has also been posed as an open question in [15]. In this note we provide a constructive solution to the problem. In particular, we adapt an idea due to A. Hatcher developed in the realm of embedding spaces and we describe an algorithm that produces an explicit isotopy between any two given projectivizations of the same affine knot. More generally, we introduce the notion of lensification of a knot in any lens space $L(p,q)$ and describe an algorithm that works in that more general setting, of which $\mathbb{R}\mathbb{P}^3\simeq L(2,1)$ is a particular instance. Finally, we apply this algorithm to several pairs of knots from the literature for which the equivalence problem was raised as an open question, finding explicit isotopies.