Twist spinning revisited

Authors:
Deborah L. Goldsmith and Louis H. Kauffman

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

MSC:
Primary 57Q45

MathSciNet review:
487047

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper contains several applications of the following theorem: The 1-twist spin of any knot is interchangeable with the standard unknotted -sphere *K* in by means of a homeomorphism of triples which reverses the orientation of , and preserves the orientations of *K* and . 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 is fiber equivalent to one of or where .

**[1]**Alan Durfee and Louis Kauffman,*Periodicity of branched cyclic covers*, Math. Ann.**218**(1975), no. 2, 157–174. MR**0385872****[2]**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****[3]**R. H. Fox,*Some 𝑛-dimensional manifolds that have the same fundamental group.*, Michigan Math. J.**15**(1968), 187–189. MR**0232378****[4]**Deborah L. Goldsmith,*Symmetric fibered links*, Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox), Princeton Univ. Press, Princeton, N.J., 1975, pp. 3–23. Ann. of Math. Studies, No. 84. MR**0380766****[5]**D. Roseman,*Half-twist spinning of arc pairs and inversions of knots*, Illinois J. Math. (to appear).**[6]**E. C. Zeeman,*Twisting spun knots*, Trans. Amer. Math. Soc.**115**(1965), 471–495. MR**0195085**, 10.1090/S0002-9947-1965-0195085-8

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

Keywords:
Manifold,
knot,
book structure,
twist spinning

Article copyright:
© Copyright 1978
American Mathematical Society