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

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 filtere...

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Bibliographic Details
Published in:Topology and its applications 2019-09, Vol.264, p.210-222
Main Authors: Ito, Noboru, Sakurai, Migiwa
Format: Article
Language:English
Subjects:
Finite type invariants
Forbidden moves
Knots
Unknotting operations
Virtual knots
Virtualizations
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Summary: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.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2019.06.019