High-dimensional knots corresponding to the fractional Fibonacci groups

Andrzej Szczepański; Andreĭ Vesnin

Fundamenta Mathematicae (1999)

  • Volume: 161, Issue: 1-2, page 235-240
  • ISSN: 0016-2736

Abstract

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We prove that the natural HNN-extensions of the fractional Fibonacci groups are the fundamental groups of high-dimensional knot complements. We also give some characterization and interpretation of these knots. In particular we show that some of them are 2-knots.

How to cite

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Szczepański, Andrzej, and Vesnin, Andreĭ. "High-dimensional knots corresponding to the fractional Fibonacci groups." Fundamenta Mathematicae 161.1-2 (1999): 235-240. <http://eudml.org/doc/212403>.

@article{Szczepański1999,
abstract = {We prove that the natural HNN-extensions of the fractional Fibonacci groups are the fundamental groups of high-dimensional knot complements. We also give some characterization and interpretation of these knots. In particular we show that some of them are 2-knots.},
author = {Szczepański, Andrzej, Vesnin, Andreĭ},
journal = {Fundamenta Mathematicae},
keywords = {fiber 2-knot; HNN-extension},
language = {eng},
number = {1-2},
pages = {235-240},
title = {High-dimensional knots corresponding to the fractional Fibonacci groups},
url = {http://eudml.org/doc/212403},
volume = {161},
year = {1999},
}

TY - JOUR
AU - Szczepański, Andrzej
AU - Vesnin, Andreĭ
TI - High-dimensional knots corresponding to the fractional Fibonacci groups
JO - Fundamenta Mathematicae
PY - 1999
VL - 161
IS - 1-2
SP - 235
EP - 240
AB - We prove that the natural HNN-extensions of the fractional Fibonacci groups are the fundamental groups of high-dimensional knot complements. We also give some characterization and interpretation of these knots. In particular we show that some of them are 2-knots.
LA - eng
KW - fiber 2-knot; HNN-extension
UR - http://eudml.org/doc/212403
ER -

References

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  1. [1] K. Brown, Cohomology of Groups, Springer, New York, 1982. Zbl0584.20036
  2. [2] H. Helling, A. Kim and J. Mennicke, A geometric study of Fibonacci groups, J. Lie Theory 8 (1998), 1-23. Zbl0896.20026
  3. [3] J. Hillman, Abelian normal subgroups of two-knot groups, Comment. Math. Helv. 61 (1986), 122-148. Zbl0611.57015
  4. [4] J. Hillman, 2-Knots and Their Groups, Austral. Math. Soc. Lecture Ser. 5, 1989. 
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  6. [6] A. Kim and A. Vesnin, A topological study of the fractional Fibonacci groups, Siberian Math. J. 39 (1998). 
  7. [7] C. MacLachlan, Generalizations of Fibonacci numbers, groups and manifolds, in: Combinatorial and Geometric Group Theory (Edinburgh, 1993), A. J. Duncan, N. D. Gilbert and J. Howie (eds.), London Math. Soc. Lecture Note Ser. 204, Cambridge Univ. Press, 1995, 233-238. Zbl0851.20026
  8. [8] W. Magnus, A. Karras and D. Solitar, Combinatorial Group Theory, Wiley Interscience, New York, 1966. 
  9. [9] S. Plotnik, Equivariant intersection forms, knots in , and rotations in 2-spheres, Trans. Amer. Math. Soc. 296 (1986), 543-575. 
  10. [10] D. Rolfsen, Knots and Links, Publish or Perish, Berkeley, CA, 1976. 
  11. [11] A. Szczepański, High dimensional knot groups and HNN extensions of the Fibonacci groups, J. Knot Theory Ramifications 7 (1998), 503-508. Zbl0908.57005
  12. [12] C. Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965), 471-495. Zbl0134.42902

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