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

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 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**On n-trivialities of classical and virtual knots for some unknotting operations.** / Ito, Noboru; Katou, Migiwa.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - On n-trivialities of classical and virtual knots for some unknotting operations

AU - Ito, Noboru

AU - Katou, Migiwa

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.).

AB - 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.).

KW - Finite type invariants

KW - Forbidden moves

KW - Knots

KW - Unknotting operations

KW - Virtu-alizations

KW - Virtual knots

UR - http://www.scopus.com/inward/record.url?scp=85060711520&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85060711520&partnerID=8YFLogxK

U2 - 10.2969/jmsj/77787778

DO - 10.2969/jmsj/77787778

M3 - Article

AN - SCOPUS:85060711520

VL - 71

SP - 329

EP - 347

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

ER -