## Abstract

Satoh and Taniguchi introduced the n-writhe Jn for each non-zero integer n, which is an integer invariant for virtual knots. The sequence of n-writhes {Jn}_{n}≠_{0} of a virtual knot K satisfies ^{∑}_{n}≠_{0} nJn(K) = 0. They showed that for any sequence of integers {cn}_{n}≠_{0} with ^{∑}_{n}≠_{0} ncn = 0, there exists a virtual knot K with Jn(K) = cn for any n ≠ 0. It is obvious that the virtualization of a real crossing is an unknotting operation for virtual knots. The unknotting number by the virtualization is called the virtual unknotting number and is denoted by u^{v}. In this paper, we show that if {cn}_{n}≠0 is a sequence of integers with ^{∑}_{n}≠_{0} ncn = 0, then there exists a virtual knot K such that u^{v}(K) = 1 and Jn(K) = cn for any n ≠ 0.

Original language | English |
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Pages (from-to) | 983 |

Number of pages | 1 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 73 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2021 |

## Keywords

- Gauss diagram
- N-writhe
- Virtual knot
- Virtualization

## ASJC Scopus subject areas

- Mathematics(all)