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

Noboru Ito, Migiwa Katou

Research output: Contribution to journalArticle

1 Citation (Scopus)

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
Pages (from-to)329-347
Number of pages19
JournalJournal of the Mathematical Society of Japan
Volume71
DOIs
Publication statusPublished - 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.

In: Journal of the Mathematical Society of Japan, Vol. 71, 01.01.2019, p. 329-347.

Research output: Contribution to journalArticle

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