On \(n\)-trivialities of classical and virtual knots for some unknotting operations. (English) Zbl 1426.57035

Summary: 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 M. Goussarov et al. [Topology 39, No. 5, 1045–1068 (2000; Zbl 1006.57005)] 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.).


57K16 Finite-type and quantum invariants, topological quantum field theories (TQFT)
57K12 Generalized knots (virtual knots, welded knots, quandles, etc.)


Zbl 1006.57005
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