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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Twist spinning revisited
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by Deborah L. Goldsmith and Louis H. Kauffman PDF
Trans. Amer. Math. Soc. 239 (1978), 229-251 Request permission

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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Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 239 (1978), 229-251
  • MSC: Primary 57Q45
  • DOI: https://doi.org/10.1090/S0002-9947-1978-0487047-7
  • MathSciNet review: 487047