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 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 $\le n+1$ is strictly stronger than that of $F$-order $\le n$ and that of GPV-order $\le 2n+1$. To obtain the result, we show that the set of virtual knot invariants of $F$-order $\le n$ contains every Goussarov-Polyak-Viro invariant of GPV-order $\le 2n+1$, which implies that the set of virtual knot invariants of $F$-order is a complete invariant of classical and virtual knots.