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On Gusarov's groups of knots

Published online by Cambridge University Press:  01 January 1999

KA YI NG  and
TED STANFORD
KA YI NG
Affiliation:
Wall Street Systems, 30 Broad Street, New York, NY 10004
TED STANFORD
Affiliation:
Department of Mathematics, United States Naval Academy, Annapolis, MD 21402; e-mail: [email protected]
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Abstract

We give a construction of Gusarov's groups [Gscr ]n of knots based on pure braid commutators, and show that any element of [Gscr ]n is represented by an infinite number of prime alternating knots of braid index less than or equal to n+1. We also study [Vscr ]n, the torsion-free part of [Gscr ]n, which is the group of equivalence classes of knots which cannot be distinguished by any rational Vassiliev invariant of order less than or equal to n. Generalizing the Gusarov–Ohyama definition of n-triviality, we give a characterization of the elements of the nth group of the lower central series of an arbitrary group.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 126 , Issue 1 , January 1999 , pp. 63 - 76
Copyright
The Cambridge Philosophical Society 1999

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