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Alexander polynomials of closed 3-braids

Published online by Cambridge University Press:  24 October 2008

H. R. Morton
H. R. Morton
Affiliation:
Department of Pure Mathematics, University of Liverpool
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Extract

The knots and links which can arise as the closure of 3-string braids, and their relations to the braids which give rise to them have been studied by Murasugi [5] and others, including Hartley [2] and more recently Przytycki[6]. Three-braids appear to form a rather special class among braids from some points of view [3]; they are also the only group of braids for which Burau's representation is known to be faithful [1]. They are, however, varied enough to provide an interesting range of knots and links on which to test a number of conjectures.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 2 , September 1984 , pp. 295 - 299
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

REFERENCES

[1]Birman, J. S.. Braids, Links and Mapping-Class Groups. Annals of Maths. Studies 82 (Princeton University Press, 1974).Google Scholar
[2]Hartley, R.. On the classification of three-braid links. Abh. Math. Sem. Univ. Hamburg 50 (1980), 108–117.CrossRefGoogle Scholar
[3]Morton, H. R.. Closed braids which are not prime knots. Math. Proc. Cambridge Philos. Soc. 86 (1979), 421–426.CrossRefGoogle Scholar
[4]Morton, H. R.. Exchangeable braids. To appear in Proceedings of Sussex Conference 1982.Google Scholar
[5]Murasugi, K.. On Closed 3-braids. Mem. Amer. Math. Soc. 151 (1974).Google Scholar
[6]Przytycki, J.. Incompressibility of surfaces in 3-manifolds. Ph.D. thesis, Columbia University (1981).Google Scholar