Deforming twist-spun knots

Litherland, R. A.

doi:10.1090/S0002-9947-1979-0530058-4

Deforming twist-spun knots
HTML articles powered by AMS MathViewer

by R. A. Litherland PDF

Trans. Amer. Math. Soc. 250 (1979), 311-331 Request permission

Abstract:

In [15] Zeeman introduced the process of twist-spinning an n-knot to obtain an (n + l)-knot, and proved the remarkable theorem that a twist-spun knot is fibred. In [2] Fox described another deformation which can be applied during the spinning process, and which he called rolling. We show that, provided one combines the rolling with a twist, the resulting knot is again fibred. In fact, this result holds for a larger class of deformations, defined below.

References

Marshall M. Cohen, A general theory of relative regular neighborhoods, Trans. Amer. Math. Soc. 136 (1969), 189–229. MR 248802, DOI 10.1090/S0002-9947-1969-0248802-6
R. H. Fox, Rolling, Bull. Amer. Math. Soc. 72 (1966), 162–164. MR 184221, DOI 10.1090/S0002-9904-1966-11467-2

Homeotopy groups of fibred knots and links

13

Herman Gluck, The embedding of two-spheres in the four-sphere, Trans. Amer. Math. Soc. 104 (1962), 308–333. MR 146807, DOI 10.1090/S0002-9947-1962-0146807-0
Deborah Louise Goldsmith, Motions of links in the $3$-sphere, Bull. Amer. Math. Soc. 80 (1974), 62–66. MR 328914, DOI 10.1090/S0002-9904-1974-13353-7
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
C. McA. Gordon, Some higher-dimensional knots with the same homotopy groups, Quart. J. Math. Oxford Ser. (2) 24 (1973), 411–422. MR 326746, DOI 10.1093/qmath/24.1.411
Mitsuyoshi Kato, A concordance classification of $\textrm {PL}$ homeomorphisms of $S^{p}\times S^{q}$, Topology 8 (1969), 371–383. MR 256401, DOI 10.1016/0040-9383(69)90023-8
John Milnor, On the $3$-dimensional Brieskorn manifolds $M(p,q,r)$, 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. 175–225. MR 0418127
Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
Jonathan Simon, Roots and centralizers of peripheral elements in knot groups, Math. Ann. 222 (1976), no. 3, 205–209. MR 418079, DOI 10.1007/BF01362577
Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594
C. T. C. Wall, Locally flat $\textrm {PL}$ submanifolds with codimension two, Proc. Cambridge Philos. Soc. 63 (1967), 5–8. MR 227993, DOI 10.1017/s0305004100040834

Seminar on combinatorial topology

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