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Classification of knotted tori

Part of: Differential topology

Published online by Cambridge University Press:  22 January 2019

A. Skopenkov
A. Skopenkov*
Affiliation:
Moscow Institute of Physics and Technology, and Independent University of Moscow ([email protected])
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Abstract

For a smooth manifold N denote by Em(N) the set of smooth isotopy classes of smooth embeddings N → ℝm. A description of the set Em(Sp × Sq) was known only for p = q = 0 or for p = 0, m ≠ q + 2 or for 2m ⩾ 2(p + q) + max{p, q} + 4. (The description was given in terms of homotopy groups of spheres and of Stiefel manifolds.) For m ⩾ 2p + q + 3 we introduce an abelian group structure on Em(Sp × Sq) and describe this group ‘up to an extension problem’. This result has corollaries which, under stronger dimension restrictions, more explicitly describe Em(Sp × Sq). The proof is based on relations between sets Em(N) for different N and m, in particular, on a recent exact sequence of M. Skopenkov.

Keywords

embeddingisotopylinksknotted toriStiefel manifolds

MSC classification

Primary: 57R40: Embeddings
Secondary: 57R52: Isotopy
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 2 , April 2020 , pp. 549 - 567
Copyright
Copyright © Royal Society of Edinburgh 2019

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