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Unknotting of pseudo-ribbon \(n\)-knots. (English) Zbl 1052.57034

Let \(f:S^n\to \mathbb{R}^{n+2}\) be an \(n\)-knot. E. Osaga [J. Knot Theory Ramifications 10, No.1, 121-132 (2001; Zbl 1002.57051)] has shown that for dimensions \(n\geq3\) there exists an \(n\)-knot \(f:S^n\to \mathbb{R}^{n+2}\) whose projection \((\pi\circ f)(S^n)\subset \mathbb{R}^{n+1}\) is not the projection of any trivial \(n\)-knot, a behaviour that does not occur when \(n=1\). The present paper treats pseudo-ribbon \(n\)-knots for \(n\geq2\) (i.e. \(f:S^n\to \mathbb{R}^{n+2}\) such that \(\pi\circ f\) is a selftransversal immersion and the self-intersection set of \(\pi\circ f\) consists of a disjoint union of \((n-1)\)-spheres) and their following behaviour: If \(n\neq 3,4\), then the projection of any pseudo-ribbon \(n\)-knot is the projection of a trivial knot. The exeption of dimensions \(n=3,4\) is the consequence of the proof that employs the Schoenflies Theorem.

MSC:

57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)

Citations:

Zbl 1002.57051
Full Text: DOI

References:

[1] Carter J. S., Mathematical Surveys and Monographs 55, in: Knotted surfaces and their diagrams (1998) · Zbl 0904.57010
[2] DOI: 10.1090/surv/095 · doi:10.1090/surv/095
[3] Kawauchi A., Mathematics Seminar Notes 10 pp 75–
[4] DOI: 10.1142/S0218216501000767 · Zbl 1002.57051 · doi:10.1142/S0218216501000767
[5] DOI: 10.1007/978-3-642-81735-9 · doi:10.1007/978-3-642-81735-9
[6] Yajima T., Osaka J. Math. 1 pp 133–
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