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Four-genus and four-dimensional clasp number of a knot

Authors: Hitoshi Murakami and Akira Yasuhara
Journal: Proc. Amer. Math. Soc. 128 (2000), 3693-3699
MSC (2000): Primary 57M25
Published electronically: May 18, 2000
MathSciNet review: 1690998
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For a knot $K$ in the $3$-sphere, by using the linking form on the first homology group of the double branched cover of the $3$-sphere, we investigate some numerical invariants, $4$-genus $g^*(K)$, nonorientable $4$-genus $\gamma^*(K)$ and $4$-dimensional clasp number $c^*(K)$, defined from the four-dimensional viewpoint. T. Shibuya gave an inequality $g^*(K)\leq c^*(K)$, and asked whether the equality holds or not. From our result in this paper, we find that the equality does not hold in general.

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Additional Information

Hitoshi Murakami
Affiliation: Department of Mathematics, School of Science and Engineering, Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan

Akira Yasuhara
Affiliation: Department of Mathematics, Tokyo Gakugei University, Nukuikita 4-1-1, Koganei, Tokyo 184-8501, Japan
Address at time of publication: Department of Mathematics, The George Washington University, Washington, DC 20052

Keywords: 4-genus, 4-dimensional clasp number, linking form
Received by editor(s): September 29, 1998
Received by editor(s) in revised form: January 29, 1999
Published electronically: May 18, 2000
Additional Notes: The first author’s research was partially supported by Waseda University Grant for Special Research Projects (#98A-623) and Grant-in-Aid for Scientific Research (C) (#09640135), the Ministry of Education, Science, Sports and Culture.
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2000 American Mathematical Society