Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



Classifying $ 1$-handles attached to knotted surfaces

Author: Jeffrey Boyle
Journal: Trans. Amer. Math. Soc. 306 (1988), 475-487
MSC: Primary 57Q45; Secondary 57M99
MathSciNet review: 933302
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study a method of obtaining knotted surfaces in the $ 4$-sphere $ {S^4}$ by attaching embedded $ 2$-dimensional $ 1$-handles to a given knot. The main result is there is a one-to-one correspondence between the $ 1$-handles that can be attached to a knot and the double cosets of the peripheral subgroup in the group of the knot. Many examples and applications are given.

References [Enhancements On Off] (What's this?)

  • [Ar] E. Artin, Zur isotopie zweidimensionaler flachen in $ {{\mathbf{R}}^n}$, Hamburg Abh. 4 (1925), 174-177.
  • [As] K. Asano, A note on surfaces in $ 4$-spheres, Math. Sem. Notes Kobe Univ. 4 (1976), 195-198. MR 0436150 (55:9100)
  • [F] R. H. Fox, A quick trip through knot theory, Topology of $ 3$-manifolds and Related Topics, M. K. Fort, Ed., Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 120-167. MR 0140099 (25:3522)
  • [H] M. Hirsch, Notes on differential topology, Department of Math., Univ. of California, Berkeley, 1973.
  • [HK] F. Hosokawa and A. Kawaushi, Proposals for unknotted surfaces in four-spaces, Osaka J. Math. 16 (1979), 233-248. MR 527028 (81c:57018)
  • [Hu] J. F. P. Hudson, Embeddings of bounded manifolds, Proc. Cambridge Philos. Soc. 72 (1972), 11-20. MR 0298679 (45:7728)
  • [Ln] M. Lien, Construction of high dimensional knot groups from classical knot groups, Ph.D. Thesis, Univ. of Iowa, 1984.
  • [L1] R. A. Litherland, The second homology of the group of a knotted surface, Quart J. Math. Oxford (2) 32 (1981), 421-434. MR 635591 (83a:57024)
  • [L2] -, Deforming twist-spun knots, Trans. Amer. Math. Soc. 250 (1979), 11-20. MR 530058 (80i:57015)
  • [Lv] C. Livingston, Stably irreducible surfaces in $ {S^4}$, Pacific J. Math. 116 (1) (1985), 77-84. MR 769824 (86g:57024)
  • [Rf] D. Rolfsen, Knots and links, Publish or Perish, Berkely, Calif., 1976. MR 0515288 (58:24236)
  • [Ri] R. Riley, Homomorphs of knot groups on finite groups, Math. Comp. 25 (1971), 603-619. MR 0295332 (45:4399)
  • [Sc] H. Schubert, Knoten mit zwei brucken, Math. Z. 65 (1950), 133-170. MR 0082104 (18:498e)
  • [Sh] P. Shalen, Infinitely divisible elements in $ 3$-manifold groups, Knots, Groups and $ 3$-Manifolds, L. P. Newirth, Ed., Ann. of Math. Studies, no. 84, Princeton Univ. Press, Princeton, N.J., 1975, pp. 293-335. MR 0375280 (51:11476)
  • [Z] E. C. Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965), 471-495. MR 0195085 (33:3290)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57Q45, 57M99

Retrieve articles in all journals with MSC: 57Q45, 57M99

Additional Information

Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society