### 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 |
---|---|

Pages (from-to) | 210-222 |

Number of pages | 13 |

Journal | Topology and its Applications |

Volume | 264 |

DOIs | |

Publication status | Published - 2019 Sep 1 |

### Keywords

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

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

**Higher-order finite type invariants of classical and virtual knots and unknotting operations.** / Ito, Noboru; Katou, Migiwa.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Higher-order finite type invariants of classical and virtual knots and unknotting operations

AU - Ito, Noboru

AU - Katou, Migiwa

PY - 2019/9/1

Y1 - 2019/9/1

N2 - 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.

AB - 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.

KW - Finite type invariants

KW - Forbidden moves

KW - Knots

KW - Unknotting operations

KW - Virtual knots

KW - Virtualizations

UR - http://www.scopus.com/inward/record.url?scp=85067856946&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85067856946&partnerID=8YFLogxK

U2 - 10.1016/j.topol.2019.06.019

DO - 10.1016/j.topol.2019.06.019

M3 - Article

AN - SCOPUS:85067856946

VL - 264

SP - 210

EP - 222

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

ER -