Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T13:09:21.876Z Has data issue: false hasContentIssue false
  • English
  • Français

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 [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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. 

References

Aumann, R. J., ‘Asphericity of alternating knots’, Ann. of Math. (2) 64 (1956), 374–392.CrossRefGoogle Scholar
Culler, M. and Shalen, P. B., ‘Bounded separating, incompressible surfaces in knot manifolds’, Invent. Math. 75 (1984), 537–545.CrossRefGoogle Scholar
Frankl, P. and Pontrjagin, L., ‘Ein Knotensatz mit Anwendung auf die Dimensionstheorie’, Math. Ann. 102 (1930), 785–789 (in German).CrossRefGoogle Scholar
Ikeda, H., ‘Acyclic fake surfaces’, Topology 10 (1971), 9–36.CrossRefGoogle Scholar
Lyon, H. C., ‘Torus knots in the complements of links and surfaces’, Michigan Math. J. 27 (1980), 39–46.CrossRefGoogle Scholar
Neuwirth, L., ‘Interpolating manifolds for knots in S 3’, Topology 2 (1964), 359–365.CrossRefGoogle Scholar
Ozawa, M., ‘Non-triviality of generalized alternating knots’, J. Knot Theory Ramifications 15 (2006), 351–360.CrossRefGoogle Scholar
Ozawa, M., ‘Essential state surfaces for knots and links’, J. Aust. Math. Soc. 91 (2011), 391–404.CrossRefGoogle Scholar
Ozawa, M. and Rubinstein, J. H., ‘On the Neuwirth conjecture for knots’, Comm. Anal. Geom. 20 (2012), 1019–1060.CrossRefGoogle Scholar
Ozawa, M. and Tsutsumi, Y., ‘Totally knotted Seifert surfaces with accidental peripherals’, Proc. Amer. Math. Soc. 131 (2003), 3945–3954.CrossRefGoogle Scholar
Rubinstein, H., ‘Some of Hyam’s favourite problems’, in: Geometry and Topology Down Under, Contemporary Mathematics, 597 (American Mathematical Society, Providence, RI, 2013), 165–175.CrossRefGoogle Scholar
Seifert, H., ‘Über das Geschlecht von Knotten’, Math. Ann. 110 (1934), 571–592 (in German).CrossRefGoogle Scholar