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Embedding knots and links in an open book II. Bounds on arc index

Published online by Cambridge University Press:  24 October 2008

Peter R. Cromwell  and
Ian J. Nutt
Peter R. Cromwell
Affiliation:
Department of Pure Mathematics, University of Liverpool, PO Box 147, Liverpool L69 3BX
Ian J. Nutt
Affiliation:
Department of Pure Mathematics, University of Liverpool, PO Box 147, Liverpool L69 3BX
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Extract

There is an open-book decomposition of the 3-sphere which has open discs as pages and an unknotted circle as the binding. We can think of the 3-sphere as ℝ3 ∪ {∞} and of the circle as the z–axis ∪ {∞}. The pages are then half-planes Hθ at angle θ when the x–y plane has polar coordinates. In their investigation of the braid index of satellite links, Birman and Menasco [B–M] embed the companion knot in finitely many such half-planes so that the knot meets each half-plane in a single simple arc, and therefore meets the axis in a finite number of points. At the end of their paper they mention that the minimum number of planes required to present a given knot in this manner is a knot invariant and that it seems to have escaped attention. (Jósef Przytycki has since pointed out to us that the phenomenon is evident in a one-hundred-year-old paper by H. Brunn[Br].) We call this invariant the arc index of a link and denote it by α(L).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 119 , Issue 2 , February 1996 , pp. 309 - 319
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

REFERENCES

[B-J]Barnette, D. and Jukovič, E.. Hamiltonian circuits on 3-polytopes. J. Combinatorial Theory 2 (1970), 54–59.CrossRefGoogle Scholar
[B-M]Birman, J. S. and Menasco, W. W.. Special positions for essential tori in link complements. Topology, 33 (1994), 525–556.CrossRefGoogle Scholar
[Br]Brunn, H.. Über verknotete Kurven. Mathematiker-Kongresses (1898).Google Scholar
[Cr]Cromwell, P. R.. Embedding knots and links in an Open Book I: Basic properties. Topology and its Applications, to appear.Google Scholar
[Ka]Kauffman, L.. State models and the Jones polynomial. Topology 26 (1987), 395–407.CrossRefGoogle Scholar
[Mo]Morton, H. R.. Threading knot diagrams. Math. Proc. Comb. Philos. Soc. 99 (1986), 247–260.CrossRefGoogle Scholar
[Mu]Murasugi, K.. Jones polynomials and classical conjectures in knot theory. Topology 26 (1987), 187–194.CrossRefGoogle Scholar
[Nu]Nutt, I. J.. Braid index of satellite links. Ph.D. thesis (University of Liverpool, 1995).Google Scholar
[Ro]Rolfsen, D.. Knots and links (Publish or Perish Inc., 1976).Google Scholar
[Th]Thistlethwaite, M. B.. A spanning tree expansion of the Jones polynomial. Topology 26 (1987), 297–309.CrossRefGoogle Scholar
[Tu]Tutte, W. T.. A theorem on planar graphs. Trans. Amer. Math. Soc. 82 (1956), 99–116.CrossRefGoogle Scholar