Twist spinning revisited

Goldsmith, Deborah L.; Kauffman, Louis H.

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

Twist spinning revisited
HTML articles powered by AMS MathViewer

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 $g = {\text {g.c.d.}}(a,b)$.

References

Alan Durfee and Louis Kauffman, Periodicity of branched cyclic covers, Math. Ann. 218 (1975), no. 2, 157–174. MR 385872, DOI 10.1007/BF01370817
Ralph H. Fox, Covering spaces with singularities, A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, N.J., 1957, pp. 243–257. MR 0123298
R. H. Fox, Some $n$-dimensional manifolds that have the same fundamental group, Michigan Math. J. 15 (1968), 187–189. MR 232378
Deborah L. Goldsmith, Symmetric fibered links, Knots, groups, and $3$-manifolds (Papers dedicated to the memory of R. H. Fox), Ann. of Math. Studies, No. 84, Princeton Univ. Press, Princeton, N.J., 1975, pp. 3–23. MR 0380766

Half-twist spinning of arc pairs and inversions of knots

E. C. Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965), 471–495. MR 195085, DOI 10.1090/S0002-9947-1965-0195085-8