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This article is cited in 7 scientific papers (total in 7 papers)
Classification of knotted tori in 2-metastable dimension
D. Repovšab, M. B. Skopenkovcd, M. Cenceljab a University of Ljubljana
b Institute of Mathematics, Physics, and Mechanics, Ljubljana
c A. A. Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow
d King Abdullah University of Science and Technology
Abstract:
This paper is devoted to the classical Knotting Problem: for a given manifold $N$ and number $m$ describe the set of isotopy classes of embeddings $N\to S^m$. We study the specific case of knotted tori, that is, the embeddings $S^p\times S^q\to S^m$. The classification of knotted tori up to isotopy in the metastable
dimension range $m\geqslant p+\frac32q+2$, $p\leqslant q$, was given by Haefliger, Zeeman and A. Skopenkov. We consider the dimensions below the metastable range and give an explicit criterion for the finiteness of this set of isotopy classes in the 2-metastable dimension:
\medskip
Theorem
Assume that $p+\frac43q+2<m<p+\frac32q+2$ and $m>2p+q+2$. Then the set of isotopy classes of smooth embeddings $S^p\times S^q\to S^m$ is infinite if and only if either $q+1$ or $p+q+1$ is divisible by $4$.
Bibliography: 35 titles.
Keywords:
knotted torus, link, link map, embedding, surgery.
Received: 23.12.2011
Citation:
D. Repovš, M. B. Skopenkov, M. Cencelj, “Classification of knotted tori in 2-metastable dimension”, Mat. Sb., 203:11 (2012), 129–158; Sb. Math., 203:11 (2012), 1654–1681
Linking options:
https://www.mathnet.ru/eng/sm8098https://doi.org/10.1070/SM2012v203n11ABEH004281 https://www.mathnet.ru/eng/sm/v203/i11/p129
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Abstract page: | 409 | Russian version PDF: | 173 | English version PDF: | 7 | References: | 40 | First page: | 16 |
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