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COEXISTENCE OF COILED SURFACES AND SPANNING SURFACES FOR KNOTS AND LINKS

Part of: Low-dimensional topology PL-topology

Published online by Cambridge University Press:  16 July 2015

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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It is a well-known procedure for constructing a torus knot or link that first we prepare an unknotted torus and meridian disks in its complementary solid tori, and second we smooth the intersections of the boundaries of the meridian disks uniformly. Then we obtain a torus knot or link on the unknotted torus and its Seifert surface made of meridian disks. In the present paper, we generalize this procedure by using a closed fake surface and show that the two resulting surfaces obtained by smoothing triple points uniformly are essential. We also show that a knot obtained by this procedure satisfies the Neuwirth conjecture and that the distance of two boundary slopes for the knot is equal to the number of triple points of the closed fake surface.

Keywords

knotcoiled surfacespanning surfaceincompressible surfaceNeuwirth conjecture

MSC classification

Primary: 57M25: Knots and links in $S^3$
Secondary: 57Q35: Embeddings and immersions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 99 , Issue 2 , October 2015 , pp. 250 - 259
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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