### Abstract

We show that the algebraic unknotting number of a classical knot $K$, defined by Murakami [9], is equalto the minimum number of unknotting operations necessary to transform K to a knot with trivial Alexander polynomial. Furthermore, we define a new operation, called an elementary twisting operation, for smooth (2n−1)-knots with n≥1 and odd, and show that this is an unknotting operation for simple (2n−1)-knots. Moreover, the unknotting number of a simple (2n−1)-knot defined by using the elementary twisting operation isequal to the algebraic unknotting number of the S-equivalence class of its Seifert matrix ifn≥3 .

Original language | English |
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Pages (from-to) | 425-443 |

Number of pages | 19 |

Journal | Tokyo Journal of Mathematics |

Volume | 22 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 1 1999 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Tokyo Journal of Mathematics*,

*22*(2), 425-443. https://doi.org/10.3836/tjm/1270041448

**On algebraic unknotting numbers of knots.** / Saeki, Osamu.

Research output: Contribution to journal › Article

*Tokyo Journal of Mathematics*, vol. 22, no. 2, pp. 425-443. https://doi.org/10.3836/tjm/1270041448

}

TY - JOUR

T1 - On algebraic unknotting numbers of knots

AU - Saeki, Osamu

PY - 1999/1/1

Y1 - 1999/1/1

N2 - We show that the algebraic unknotting number of a classical knot $K$, defined by Murakami [9], is equalto the minimum number of unknotting operations necessary to transform K to a knot with trivial Alexander polynomial. Furthermore, we define a new operation, called an elementary twisting operation, for smooth (2n−1)-knots with n≥1 and odd, and show that this is an unknotting operation for simple (2n−1)-knots. Moreover, the unknotting number of a simple (2n−1)-knot defined by using the elementary twisting operation isequal to the algebraic unknotting number of the S-equivalence class of its Seifert matrix ifn≥3 .

AB - We show that the algebraic unknotting number of a classical knot $K$, defined by Murakami [9], is equalto the minimum number of unknotting operations necessary to transform K to a knot with trivial Alexander polynomial. Furthermore, we define a new operation, called an elementary twisting operation, for smooth (2n−1)-knots with n≥1 and odd, and show that this is an unknotting operation for simple (2n−1)-knots. Moreover, the unknotting number of a simple (2n−1)-knot defined by using the elementary twisting operation isequal to the algebraic unknotting number of the S-equivalence class of its Seifert matrix ifn≥3 .

UR - http://www.scopus.com/inward/record.url?scp=84971654546&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84971654546&partnerID=8YFLogxK

U2 - 10.3836/tjm/1270041448

DO - 10.3836/tjm/1270041448

M3 - Article

AN - SCOPUS:84971654546

VL - 22

SP - 425

EP - 443

JO - Tokyo Journal of Mathematics

JF - Tokyo Journal of Mathematics

SN - 0387-3870

IS - 2

ER -