On n-trivialities of Classical and Virtual Knots for some Unknotting Operations

Noboru Ito, Migiwa Sakurai

Research output: Contribution to journalArticlepeer-review

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 languageEnglish
JournalUnknown Journal
Publication statusPublished - 2020 Aug 6
Externally publishedYes

Keywords

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

ASJC Scopus subject areas

  • General

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