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Some groups which classify knots

Published online by Cambridge University Press:  24 October 2008

B. Zimmermann
B. Zimmermann
Affiliation:
Università degli Studi di Trieste, Dipartimento di Scienze Matematiche, 34100 Trieste, Italy
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Groups associated to a knot in S3 which classify the knot have been constructed by Conway and Gordon [4], Simon [12] and Whitten[16]; see also [1] and [19] for certain special classes of knots. In this note we show that for a knot K in S3 of the groups π1(S3/K)/〈malb〉 classify the knot; here m resp. l denote a meridian resp. longitude of the knot and 〈〉 denotes normal closure. This is based on Thurston's orbifold geometrization theorem [13–15] and the following result (see also [8, 19]):

Theorem 1. Let O1 and O2 be good closed irreducible 3-orbifolds which possess a decomposition into geometric pieces (along Euclidean 2-suborbifolds, see [2, 15]) and have infinite (orbifold-) fundamental group. Then O1 and O2 are diffeomorphic (as orbifolds) if and only if π1 O1 and π1 02 are isomorphic.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 104 , Issue 3 , November 1988 , pp. 417 - 419
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

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