### Abstract

We show that for any nontrivial knot K and any natural number n, there is a diagram D of K such that the unknotting number of D is greater than or equal to n. It is well-known that twice the unknotting number of K is less than or equal to the crossing number of K minus one. We show that the equality holds only when K is a (2, p)-torus knot.

Original language | English |
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Pages (from-to) | 1049-1063 |

Number of pages | 15 |

Journal | Journal of Knot Theory and its Ramifications |

Volume | 18 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2009 Aug |

### Fingerprint

### Keywords

- Crossing number.
- Knot
- Unknotting number
- Unknotting number of diagram

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Unknotting numbers of diagrams of a given nontrivial knot are unbounded.** / Taniyama, Kouki.

Research output: Contribution to journal › Article

*Journal of Knot Theory and its Ramifications*, vol. 18, no. 8, pp. 1049-1063. https://doi.org/10.1142/S0218216509007361

}

TY - JOUR

T1 - Unknotting numbers of diagrams of a given nontrivial knot are unbounded

AU - Taniyama, Kouki

PY - 2009/8

Y1 - 2009/8

N2 - We show that for any nontrivial knot K and any natural number n, there is a diagram D of K such that the unknotting number of D is greater than or equal to n. It is well-known that twice the unknotting number of K is less than or equal to the crossing number of K minus one. We show that the equality holds only when K is a (2, p)-torus knot.

AB - We show that for any nontrivial knot K and any natural number n, there is a diagram D of K such that the unknotting number of D is greater than or equal to n. It is well-known that twice the unknotting number of K is less than or equal to the crossing number of K minus one. We show that the equality holds only when K is a (2, p)-torus knot.

KW - Crossing number.

KW - Knot

KW - Unknotting number

KW - Unknotting number of diagram

UR - http://www.scopus.com/inward/record.url?scp=69849114529&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=69849114529&partnerID=8YFLogxK

U2 - 10.1142/S0218216509007361

DO - 10.1142/S0218216509007361

M3 - Article

VL - 18

SP - 1049

EP - 1063

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

SN - 0218-2165

IS - 8

ER -