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ESSENTIAL STATE SURFACES FOR KNOTS AND LINKS

Part of: Low-dimensional topology

Published online by Cambridge University Press:  19 March 2012

MAKOTO OZAWA
MAKOTO OZAWA*
Affiliation:
Department of Natural Sciences, Faculty of Arts and Sciences, Komazawa University, 1-23-1 Komazawa, Setagaya-ku, Tokyo, 154-8525, Japan (email: w3c@komazawa-u.ac.jp)
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Abstract

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We study a canonical spanning surface obtained from a knot or link diagram, depending on a given Kauffman state. We give a sufficient condition for the surface to be essential. By using the essential surface, we can deduce the triviality and splittability of a knot or link from its diagrams. This has been done on the extended knot or link class that includes all semiadequate, homogeneous knots and links, and most algebraic knots and links. In order to prove the main theorem, we extend Gabai’s Murasugi sum theorem to the case of nonorientable spanning surfaces.

Keywords

state surfaceknot diagramadequate knothomogeneous knotMurasugi sum

MSC classification

Secondary: 57M25: Knots and links in $S^3$
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 91 , Issue 3 , December 2011 , pp. 391 - 404
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

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