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Knots of genus one or on the number of alternating knots of given genus


Author: A. Stoimenow
Journal: Proc. Amer. Math. Soc. 129 (2001), 2141-2156
MSC (2000): Primary 57M27
DOI: https://doi.org/10.1090/S0002-9939-01-05823-3
Published electronically: February 23, 2001
MathSciNet review: 1825928
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Abstract:

We prove that any non-hyperbolic genus one knot except the trefoil does not have a minimal canonical Seifert surface and that there are only polynomially many in the crossing number positive knots of given genus or given unknotting number.


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Additional Information

A. Stoimenow
Affiliation: Ludwig-Maximilians University Munich, Mathematics Institute, Theresienstraße 39, 80333 München, Germany
Address at time of publication: Max Planck Institute of Mathematics, P.O. Box 7280, D-53072 Bonn, Germany
Email: stoimeno@informatik.hu-berlin.de, alex@mpim-bonn.mpg.de

DOI: https://doi.org/10.1090/S0002-9939-01-05823-3
Keywords: Positive knots, Seifert surfaces, Gau\ss{} diagrams, genus, unknotting number, alternating knots, Vassiliev invariants
Received by editor(s): February 11, 1999
Received by editor(s) in revised form: July 23, 1999, and October 20, 1999
Published electronically: February 23, 2001
Additional Notes: The author was supported by a DFG postdoc grant.
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2001 American Mathematical Society

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