Journal of the Mathematical Society of Japan

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

Noboru ITO and Migiwa SAKURAI

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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 $\le n-1$, 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 $\le n-1$ 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.).


The work was partially supported by Grant-in-Aid for Scientific Research (S) (No. 24224002) and by Grant for Basic Science Research Projects from The Sumitomo Foundation (No. 160556). The first author was a project researcher of Grant-in-Aid for Scientific Research (S) (2016.4–2017.3).

Article information

J. Math. Soc. Japan, Volume 71, Number 1 (2019), 329-347.

Received: 19 April 2017
Revised: 17 September 2017
First available in Project Euclid: 8 November 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

finite type invariants knots virtual knots unknotting operations virtualizations forbidden moves


ITO, Noboru; SAKURAI, Migiwa. On $n$-trivialities of classical and virtual knots for some unknotting operations. J. Math. Soc. Japan 71 (2019), no. 1, 329--347. doi:10.2969/jmsj/77787778.

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