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UNKNOTTING OF PSEUDO-RIBBON n-KNOTS

KYOUHEI YOSHIDA

Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

https://doi.org/10.1142/S0218216504003123Cited by:1 (Source: Crossref)

Abstract

Let K be a classical knot in ℝ³. We can deform the diagram of K to that of a trivial knot by changing the overcrossings and undercrossings at some double points of the diagram of K. We consider the same problem for higher dimensinal knots. Let n≥2 and π:ℝⁿ⁺²=ℝⁿ⁺¹×ℝ→ℝⁿ⁺¹ denote the natural projection map. A pseudo-ribbon n-knot is an n-knot f:Sⁿ→ℝⁿ⁺² such that the self-intersection set of π◦f:Sⁿ→ℝⁿ⁺¹ consists of only double points and is homeomorphic to a disjoint union of (n-1)-spheres. We prove that for n≠3,4, the projection (π◦f)(Sⁿ)⊂ℝⁿ⁺¹ of any pseudo-ribbon n-knot f is the projection of a trivial n-knot.

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