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Lonely knots and tangles: Identifying knots with no companions

Part of: Low-dimensional topology

Published online by Cambridge University Press:  26 February 2010

P. R. Cromwell
P. R. Cromwell
Affiliation:
School of Mathematics, University College of North Wales, Dean Street, Bangor, Gwynedd. LL57 1UT
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Extract

There are two principal ways of decomposing knots and links into simpler ones: (1) a sphere intersecting the knot in two points gives a connected sum decomposition; (2) an incompressible torus in the knot complement gives a satellite decomposition. If a knot K is such that in every connected sum decomposition one of the factors is an unknotted arc spanning the sphere then K is called a prime knot. In [L] Raymond Lickorish explored the possibility of using 2-string tangles to construct and detect prime knots. He defined prime tangles and showed that the sum of prime tangles is always a prime knot or link. Later, Quach Cam Van studied partial sums of tangles and gave necessary and sufficient conditions for the resulting tangle to be prime. In this paper, similar results are established which relate to the satellite decomposition rather than to the connected sum.

MSC classification

Secondary: 57M25: Knots and links in $S^3$
Type
Research Article
Information
Mathematika , Volume 38 , Issue 2 , December 1991 , pp. 334 - 347
Copyright
Copyright © University College London 1991

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