Vassiliev invariants of degree 2 for virtual long knots

Goussarov, Polyak, and Viro introduced finite-type invariants of virtual knots and provided two such invariants of virtual long knots of degree 2 using Gauss diagram formulas. Since any finite-type invariant is also a Vassiliev invariant, this connects Gauss diagram formulas with Vassiliev theory. Given the Gauss diagram of a virtual knot, we assign an integer degree to each chord, and by combining these degrees with the Gauss diagram formulas for degree 2 invariants, we construct two Vassiliev invariants of degree 2 for virtual long knots. Okada showed that if two knots differ by a Δ-move, then the Vassiliev invariant of degree 2 arising from the second coefficient of the Conway polynomial changes exactly by 1. As an application, by using the two Vassiliev invariants, we obtain a lower bound for the number of Δ-moves required to transform one virtual long knot diagram into another.