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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

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