Higher-order finite type invariants of classical and virtual knots and unknotting operations

Noboru Ito, Migiwa Katou

Research output: Contribution to journalArticle

Abstract

Vassiliev introduced filtered invariants of knots using an unknotting operation, called crossing changes. Goussarov, Polyak, and Viro introduced other filtered invariants of virtual knots, which order is called GPV-order, using an unknotting operation, called virtualization. We defined other filtered invariants, which order is called F-order, of virtual knots using an unknotting operation, called forbidden moves. In this paper, we show that the set of virtual knot invariants of F-order ≤n+1 is strictly stronger than that of F-order ≤n and that of GPV-order ≤2n+1. To obtain the result, we show that the set of virtual knot invariants of F-order ≤n contains every Goussarov-Polyak-Viro invariant of GPV-order ≤2n+1, which implies that the set of virtual knot invariants of F-order is a complete invariant of classical and virtual knots.

Original languageEnglish
Pages (from-to)210-222
Number of pages13
JournalTopology and its Applications
Volume264
DOIs
Publication statusPublished - 2019 Sep 1

Keywords

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

ASJC Scopus subject areas

  • Geometry and Topology

Cite this

Higher-order finite type invariants of classical and virtual knots and unknotting operations. / Ito, Noboru; Katou, Migiwa.

In: Topology and its Applications, Vol. 264, 01.09.2019, p. 210-222.

Research output: Contribution to journalArticle

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