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ArticleOriginal scientific text

Title

Knot manifolds with isomorphic spines

Authors 1, 2

Affiliations

  1. Dipartimento di Matematica, Università di Modena, via Campi 213/b, 41100 Modena, Italy
  2. Dipartimento di Matematica, Università di Milano, via Saldini 50, 20133 Milano, Italy

Abstract

We study the relation between the concept of spine and the representation of orientable bordered 3-manifolds by Heegaard diagrams. As a consequence, we show that composing invertible non-amphicheiral knots yields examples of topologically different knot manifolds with isomorphic spines. These results are related to some questions listed in [9], [11] and recover the main theorem of [10] as a corollary. Finally, an application concerning knot manifolds of composite knots with h prime factors completes the paper.

Keywords

ENG
3-manifold, spine, group presentation, Heegaard diagram, knot, knot group, knot manifold, peripheral system

Bibliography

  1. G. Burde and H. Zieschang, Knots, Walter de Gruyter, Berlin, 1985.
  2. A. Cavicchioli, Imbeddings of polyhedra in 3-manifolds, Ann. Mat. Pura Appl. 162 (1992), 157-177.
  3. R. Craggs, Free Heegaard diagrams and extended Nielsen transformations, I, Michigan Math. J. 26 (1979), 161-186; II, Illinois J. Math. 23 (1979), 101-127.
  4. M. Culler, C. Mc A. Gordon, J. Luecke and P. Shalen, Dehn surgery on knots, Bull. Amer. Math. Soc. 13 (1985), 43-45; Ann. of Math. 125 (1987), 237-300.
  5. C. D. Feustel and W. Whitten, Groups and complement of knots, Canad. J. Math. 30 (1978), 1284-1295.
  6. C. Mc A. Gordon and J. Luecke, Knots are determined by their complements, Bull. Amer. Math. Soc. 20 (1989), 83-87; J. Amer. Math. Soc. 2 (1989), 371-415.
  7. J. Hempel, 3-manifolds, Princeton Univ. Press, Princeton, N.J., 1976.
  8. L. H. Kauffman, On knots, Ann. of Math. Stud. 115, Princeton Univ. Press, Princeton, N.J., 1987.
  9. R. Kirby, Problems in low dimensional manifold theory, in: Proc. Sympos. Pure Math. 32, Amer. Math. Soc., Providence, R.I., 1978, 273-312.
  10. W. J. R. Mitchell, J. Przytycki and D. Repovš, On spines of knot spaces, Bull. Polish Acad. Sci. 37 (1989), 563-565.
  11. D. Repovš, Regular neighbourhoods of homotopically PL embedded compacta in 3-manifolds, Suppl. Rend. Circ. Mat. Palermo 18 (1988), 415-422.
  12. D. Rolfsen, Knots and Links, Math. Lecture Ser. 7, Publish or Perish, Berkeley, 1976.
  13. T. B. Rushing, Topological Embeddings, Academic Press, New York, 1973.
  14. J. Simon, On the problems of determining knots by their complements and knot complements by their groups, Proc. Amer. Math. Soc. 57 (1976), 140-142.
  15. J. Simon, How many knots have the same group?, ibid. 80 (1980), 162-166.
  16. J. Singer, Three-dimensional manifolds and their Heegaard diagrams, Trans. Amer. Math. Soc. 35 (1933), 88-111.
  17. J. Stillwell, Classical Topology and Combinatorial Group Theory, Springer, New York, 1980.
  18. F. Waldhausen, On irreducible 3-manifolds which are sufficiently large, Ann. of Math. 87 (1968), 56-88.
  19. W. Whitten, Rigidity among prime-knot complements, Bull. Amer. Math. Soc. 14 (1986), 299-300.
  20. W. Whitten, Knot complements and groups, Topology 26 (1987), 41-44.
  21. P. Wright, Group presentations and formal deformations, Trans. Amer. Math. Soc. 208 (1975), 161-169.
  22. E. C. Zeeman, Seminar on Combinatorial Topology, mimeographed notes, Inst. des Hautes Études Sci., Paris, 1963.

Additional information

http://matwbn.icm.edu.pl/ksiazki/fm/fm145/fm14515.pdf

Fundamenta Mathematicae
1994/145/1
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Pages:
79-89
Main language of publication
English
Received
1993-06-17
Published
1994
Exact and natural sciences