### Abstract

In this paper, we introduce a new nontrivial filtration, called F-order, for classical and virtual knot invariants; this filtration produces filtered knot invariants, which are called finite type invariants similar to Vassiliev knot invariants. Finite type invariants introduced by Goussarov, Polyak, and Viro are well-known, and we call them finite type invariants of GPV-order. We show that for any positive integer n and for any classical knot K, there exist infinitely many of nontrivial classical knots, all of whose finite type invariants of GPV-order ≤ n − 1, coincide with those of K (Theorem 1). Further, we show that for any positive integer n, there exists a nontrivial virtual knot whose finite type invariants of our F-order ≤ n − 1 coincide with those of the trivial knot (Theorem 2). In order to prove Theorem 1 (Theorem 2, resp.), we define an n-triviality via a certain unknotting operation, called virtualization (forbidden moves, resp.), and for any positive integer n, find an n-trivial classical knot (virtual knot, resp.).

Original language | English |
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Pages (from-to) | 329-347 |

Number of pages | 19 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 71 |

DOIs | |

Publication status | Published - 2019 Jan 1 |

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### Keywords

- Finite type invariants
- Forbidden moves
- Knots
- Unknotting operations
- Virtu-alizations
- Virtual knots

### ASJC Scopus subject areas

- Mathematics(all)