### Abstract

Vassiliev introduced filtered invariants of knots using an unknotting operation, called crossing changes. Goussarov, Polyak, and Viro introduced other filtered invariants of virtual knots, which order is called GPV-order, using an unknotting operation, called virtualization. We defined other filtered invariants, which order is called F-order, of virtual knots using an unknotting operation, called forbidden moves. In this paper, we show that the set of virtual knot invariants of F-order ≤n+1 is strictly stronger than that of F-order ≤n and that of GPV-order ≤2n+1. To obtain the result, we show that the set of virtual knot invariants of F-order ≤n contains every Goussarov-Polyak-Viro invariant of GPV-order ≤2n+1, which implies that the set of virtual knot invariants of F-order is a complete invariant of classical and virtual knots.

Original language | English |
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Pages (from-to) | 210-222 |

Number of pages | 13 |

Journal | Topology and its Applications |

Volume | 264 |

DOIs | |

Publication status | Published - 2019 Sep 1 |

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### Keywords

- Finite type invariants
- Forbidden moves
- Knots
- Unknotting operations
- Virtual knots
- Virtualizations

### ASJC Scopus subject areas

- Geometry and Topology