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

TL;DR

This paper introduces a new filtration called F-order for classical and virtual knots, producing finite type invariants that help distinguish knots and analyze their triviality under specific unknotting operations.

Contribution

It defines the F-order filtration for knot invariants and demonstrates the existence of infinitely many knots sharing invariants with trivial or given knots under this filtration.

Findings

Existence of infinitely many classical knots with identical finite type invariants of GPV-order < n as a given knot.

Existence of nontrivial virtual knots with invariants matching the trivial knot for F-order < n.

Introduction of n-triviality via virtualization and forbidden moves for classical and virtual knots.

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 $< n$, 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 $< n$ 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.).