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Some infinite families of satellite knots with given Alexander polynomial

Part of: Low-dimensional topology

Published online by Cambridge University Press:  26 February 2010

P. R. Cromwell
P. R. Cromwell
Affiliation:
Dr. P. R. Cromwell, School of Mathematics, University of Wales, Bangor, Gwynedd, LL57 1UT.
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Extract

For satellite knots there is a well-known formula which relates the Alexander polynomial of the satellite to those of a companion knot and the corresponding pattern. If &s, &C and &P are the Alexander polynomials of a satellite, companion and pattern respectively then

where is the linking number of P with a meridian of the companion torus (see [BZ], p. 118). Analogous relationships do not exist for other knot polynomials [MS]. This suggests that the existence of the above formula depends more on the geometry underlying the polynomial than on the geometry of the satellite construction.

MSC classification

Secondary: 57M25: Knots and links in $S^3$

Information

Type
Research Article
Information
Mathematika , Volume 38 , Issue 1 , June 1991 , pp. 156 - 169
Copyright
Copyright © University College London 1991

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