### Abstract

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 γ*(K) and 4-dimensional clasp number c* (K), defined from the four-dimensional viewpoint. T. Shibuya gave an inequality g*(K) ≤ 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.

Original language | English |
---|---|

Pages (from-to) | 3693-3699 |

Number of pages | 7 |

Journal | Proceedings of the American Mathematical Society |

Volume | 128 |

Issue number | 12 |

Publication status | Published - 2000 Dec 1 |

### Fingerprint

### Keywords

- 4-dimensional clasp number
- 4-genus
- Linking form

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*128*(12), 3693-3699.

**Four-genus and four-dimensional clasp number of a knot.** / Murakami, Hitoshi; Yasuhara, Akira.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 128, no. 12, pp. 3693-3699.

}

TY - JOUR

T1 - Four-genus and four-dimensional clasp number of a knot

AU - Murakami, Hitoshi

AU - Yasuhara, Akira

PY - 2000/12/1

Y1 - 2000/12/1

N2 - 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 γ*(K) and 4-dimensional clasp number c* (K), defined from the four-dimensional viewpoint. T. Shibuya gave an inequality g*(K) ≤ 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.

AB - 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 γ*(K) and 4-dimensional clasp number c* (K), defined from the four-dimensional viewpoint. T. Shibuya gave an inequality g*(K) ≤ 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.

KW - 4-dimensional clasp number

KW - 4-genus

KW - Linking form

UR - http://www.scopus.com/inward/record.url?scp=23044521880&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23044521880&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:23044521880

VL - 128

SP - 3693

EP - 3699

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 12

ER -