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

top
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

top

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

top
  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. 
  5. [5] D. Johnson, Presentation of Groups, London Math. Soc. Lecture Note Ser. 22, Cambridge Univ. Press, 1976. 
  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 S 4 , 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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.