Twist spinning revisited

Goldsmith, Deborah L.; Kauffman, Louis H.

doi:10.1090/S0002-9947-1978-0487047-7

Twist spinning revisited
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by Deborah L. Goldsmith and Louis H. Kauffman

Trans. Amer. Math. Soc. 239 (1978), 229-251

DOI: https://doi.org/10.1090/S0002-9947-1978-0487047-7

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Abstract:

This paper contains several applications of the following theorem: The 1-twist spin ${L_1}(k)$ of any knot $k \subset {S^{n - 1}}$ is interchangeable with the standard unknotted $(n - 2)$-sphere K in ${S^n}$ by means of a homeomorphism of triples $h:({S^n},K,{L_1}(k)) \to ({S^n},{L_1}(k),K)$ which reverses the orientation of ${S^n}$, and preserves the orientations of K and ${L_1}(k)$. One of these applications is Zeeman’s Theorem about twist spun knots; another is a proof of a conjecture of R. H. Fox about certain manifolds which have the same fundamental group. We also prove that the iterated twist spun knot ${L_{a,b}}(k) \subset {S^{n + 1}}$ is fiber equivalent to one of ${L_{0,g}}(k)$ or ${L_{g,g}}(k)$ where $g = {\text {g.c.d.}}(a,b)$.

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