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Knot projections and Coxeter groups

Part of: Low-dimensional topology

Published online by Cambridge University Press:  09 April 2009

A. M. Brunner  and
Y. W. Lee
A. M. Brunner
Affiliation:
Department of Mathematics, University of Wisconsin–Parkside, Kenosha, Wisconsin 53141, U.S.A.
Y. W. Lee
Affiliation:
Department of Mathematics, University of Wisconsin–Parkside, Kenosha, Wisconsin 53141, U.S.A.
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Abstract

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Every knot admits a special projection with the property that under the projection discs in the canonical Seifert surface project disjointly. Under an isotopy, such a projection can be turned into a connected sum of what we call inseparable projections. The main result is that if there is no band in an inseparable projection with half-twisting number +1 or −1, then the projection is not a projection of the trivial knot. To prove this a non-cyclic Coxeter group is constructed as a quotient of the knot group. The construction is possibly of interest in itself. The techniques developed are applied to give a criterion to decide when an inseparable projection with 3 discs comes from the trivial knot.

MSC classification

Secondary: 57M25: Knots and links in $S^3$
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 56 , Issue 1 , February 1994 , pp. 1 - 16
Copyright
Copyright © Australian Mathematical Society 1994

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