  Although it is known that the dimension of the Vassiliev invariants of degree
three of long virtual knots is seven, the complete list of seven distinct Gauss
diagram formulas have been unknown explicitly, where only one known formula was
revised without proof. In this paper, we give seven Gauss diagram formulas to
present the seven invariants of the degree three (Proposition 4). We further
give 23 Gauss diagram formulas of classical knots (Proposition 5). In
particular, the Polyak-Viro Gauss diagram formula [19] is not a long virtual
knot invariant; however, it is included in the list of 23 formulas. It has been
unknown whether this formula would be available by arrow diagram calculus
automatically. In consequence, as it relates to the conjecture of
Goussarov-Polyak-Viro [8, Conjecture 3.C], for all the degree three finite type
long virtual knot invariants, each Gauss diagram formula is represented as
those of Vassiliev invariants of classical knots (Theorem 1).

Geometric Topology;Algebraic Topology