Knot manifolds with isomorphic spines

Alberto Cavicchioli; Friedrich Hegenbarth

Fundamenta Mathematicae (1994)

  • Volume: 145, Issue: 1, page 79-89
  • ISSN: 0016-2736

Abstract

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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.

How to cite

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Cavicchioli, Alberto, and Hegenbarth, Friedrich. "Knot manifolds with isomorphic spines." Fundamenta Mathematicae 145.1 (1994): 79-89. <http://eudml.org/doc/212035>.

@article{Cavicchioli1994,
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.},
author = {Cavicchioli, Alberto, Hegenbarth, Friedrich},
journal = {Fundamenta Mathematicae},
keywords = {3-manifold; spine; group presentation; Heegaard diagram; knot; knot group; knot manifold; peripheral system; 3-manifolds; Heegaard diagrams; invertible non-amphicheiral knots; knot manifolds with isomorphic spines; composite knots},
language = {eng},
number = {1},
pages = {79-89},
title = {Knot manifolds with isomorphic spines},
url = {http://eudml.org/doc/212035},
volume = {145},
year = {1994},
}

TY - JOUR
AU - Cavicchioli, Alberto
AU - Hegenbarth, Friedrich
TI - Knot manifolds with isomorphic spines
JO - Fundamenta Mathematicae
PY - 1994
VL - 145
IS - 1
SP - 79
EP - 89
AB - 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.
LA - eng
KW - 3-manifold; spine; group presentation; Heegaard diagram; knot; knot group; knot manifold; peripheral system; 3-manifolds; Heegaard diagrams; invertible non-amphicheiral knots; knot manifolds with isomorphic spines; composite knots
UR - http://eudml.org/doc/212035
ER -

References

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  1. [1] G. Burde and H. Zieschang, Knots, Walter de Gruyter, Berlin, 1985. 
  2. [2] A. Cavicchioli, Imbeddings of polyhedra in 3-manifolds, Ann. Mat. Pura Appl. 162 (1992), 157-177. Zbl0777.57003
  3. [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. Zbl0441.57011
  4. [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. Zbl0571.57008
  5. [5] C. D. Feustel and W. Whitten, Groups and complement of knots, Canad. J. Math. 30 (1978), 1284-1295. Zbl0373.55003
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  9. [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. [10] W. J. R. Mitchell, J. Przytycki and D. Repovš, On spines of knot spaces, Bull. Polish Acad. Sci. 37 (1989), 563-565. Zbl0758.57006
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  15. [15] J. Simon, How many knots have the same group?, ibid. 80 (1980), 162-166. Zbl0447.57004
  16. [16] J. Singer, Three-dimensional manifolds and their Heegaard diagrams, Trans. Amer. Math. Soc. 35 (1933), 88-111. Zbl0006.18501
  17. [17] J. Stillwell, Classical Topology and Combinatorial Group Theory, Springer, New York, 1980. 
  18. [18] F. Waldhausen, On irreducible 3-manifolds which are sufficiently large, Ann. of Math. 87 (1968), 56-88. Zbl0157.30603
  19. [19] W. Whitten, Rigidity among prime-knot complements, Bull. Amer. Math. Soc. 14 (1986), 299-300. Zbl0602.57003
  20. [20] W. Whitten, Knot complements and groups, Topology 26 (1987), 41-44. Zbl0607.57004
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  22. [22] E. C. Zeeman, Seminar on Combinatorial Topology, mimeographed notes, Inst. des Hautes Études Sci., Paris, 1963. 

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