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Transactions of the American Mathematical Society

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Construction by isotopy. II

Author: Daniel S. Silver
Journal: Trans. Amer. Math. Soc. 317 (1990), 813-823
MSC: Primary 57Q45; Secondary 57M25
MathSciNet review: 987168
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Abstract: Construction by isotopy is a technique introduced by Iain R. Aitchison for obtaining doubly slice fibered knots in any dimension. We show that if $ k$ is any doubly slice fibered $ (n - 2)$-knot, $ n \geqslant 5$, such that $ {\pi _1}({S^n} - k) \cong Z$, then $ k$ is constructible by isotopy. We also prove that the $ m$-twist-spin of any doubly slice knot is constructible by isotopy. Consequently, there exists a double slice knot constructible by isotopy that is not the double of any disk knot. We also give an example of a doubly slice fibered $ 6$-knot that is not constructible by isotopy.

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  • [A] I. R. Aitchison, Isotoping and twisting knots and ribbons, Ph.D. Dissertation, University of California at Berkeley, 1984.
  • [Ba] H. Bass, The stable structure of linear groups, Bull. Amer. Math. Soc. 70 (1964), 429-433. MR 0160825 (28:4035)
  • [BrLe] W. Browder and J. Levine, Fibering manifolds over a circle, Comment. Math. Helv. 40 (1965), 153-160. MR 0195104 (33:3309)
  • [CaR] S. Cappell and D. Ruberman, Imbeddings and homology cobordisms of lens spaces, Comment. Math. Helv. 63 (1988), 75-88. MR 928028 (89c:57040)
  • [Ce] J. Cerf, La stratification naturelle des espaces de fonctions différentiables reeles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math., no. 39. MR 0292089 (45:1176)
  • [Co] M. Cohen, A course in simple-homotopy theory, Springer-Verlag, 1973. MR 0362320 (50:14762)
  • [F] F. T. Farrell, The obstruction to fibering a manifold over a circle, Indiana Univ. Math. J. 21 (1971), 315-346. MR 0290397 (44:7578)
  • [HiSil] L. R. Hitt and D. Silver, Families of ribbon knots via Stallings twists, Preprint.
  • [HoT] F. Hosokawa and H. Terasaka, On the unknotted sphere $ {S^2}$ in $ {E^4}$, Osaka J. Math. 13 (1961), 265-270. MR 0142117 (25:5510)
  • [Le] J. Levine, Doubly slice knots and doubled disk knots, Michigan Math. J. 30 (1983), 249-256. MR 718271 (85h:57024)
  • [Li] R. A. Litherland, Deforming twist-spun knots, Trans. Amer. Math. Soc. 250 (1979), 311-331. MR 530058 (80i:57015)
  • [Mi] J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358-426. MR 0196736 (33:4922)
  • [Mu] J. R. Munkres, Elementary differential topology, Ann. of Math. Stud., no. 54, Princeton Univ. Press, Princeton, N. J., 1966. MR 0198479 (33:6637)
  • [R$ _{1}$] D. Ruberman, Imbedding punctured lens spaces and connected sums, Pacific J. Math. 113 (1984), 481-491. MR 749551 (85j:57045)
  • [R$ _{2}$] -, The Casson-Gordon invariants in high-dimensional knot theory, Trans. Amer. Math. Soc. 306 (1988), 579-595. MR 933307 (89g:57031)
  • [Sie] L. C. Siebenmann, A total Whitehead obstruction to fibering over a circle, Comment. Math. Helv. 45 (1970), 1-48. MR 0287564 (44:4768)
  • [Sil] D. Silver, On Aitchison's construction by isotopy, Trans. Amer. Math. Soc. 305 (1988), 641-652. MR 924773 (89d:57030)
  • [Sm] L. Smolinsky, Double disk knots and a link invariant, Ph. D. Dissertation, Brandeis Univ., Waltham, Mass., 1985.
  • [Sp] E. H. Spanier, Algebraic topology, McGraw-Hill, 1966. MR 0210112 (35:1007)
  • [St] J. Stallings, On infinite processes leading to differentiability in the complement of a point, Differential and Combinatorial Topology (A Symposium in honor of M. Morse), Princeton Univ. Press, Princeton, N. J., 1965, pp. 245-254. MR 0180983 (31:5213)
  • [Z] E. C. Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965), 471-495. MR 0195085 (33:3290)

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