Classification of knotted tori

TL;DR

This paper introduces a new algebraic structure to classify smooth embeddings of knotted tori in Euclidean space, extending previous results and providing explicit descriptions under certain dimension restrictions.

Contribution

It develops an abelian group structure on the set of embeddings of knotted tori and describes this structure up to an extension problem, advancing the classification of such embeddings.

Findings

Defined an abelian group structure on $E^m(S^p\times S^q)$ for certain dimensions.

Provided explicit descriptions of the embedding set under stronger dimension restrictions.

Connected the classification problem to homotopy groups and recent exact sequences.

Abstract

For a smooth manifold $N$ denote by $E^m(N)$ the set of smooth isotopy classes of smooth embeddings $N\to\mathbb R^m$. A description of the set $E^m(S^p\times S^q)$ was known only for $p=q=0$ or for $p=0$, $m\ne q+2$ or for $2m\ge 2(p+q)+\max\{p,q\}+4$. (The description was given in terms of homotopy groups of spheres and of Stiefel manifolds.) For $m\ge2p+q+3$ we introduce an abelian group structure on $E^m(S^p\times S^q)$ and describe this group `up to an extension problem'. This result has corollaries which, under stronger dimension restrictions, more explicitly describe $E^m(S^p\times S^q)$. The proof is based on relations between sets $E^m(N)$ for different $N$ and $m$, in particular, on a recent exact sequence of M. Skopenkov.