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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 has 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 2 3 Gauss diagram formulas of classical knots (Proposition 5). In particular, the Polyak–Viro Gauss diagram formula [M. Polyak and O. Viro, Gauss diagram formulas for Vassiliev invariants, Int. Math. Res. Not.1994 (1994) 445–453] is not a long virtual knot invariant; however, it is included in the list of 2 3 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 [Finite-type invariants of classical and virtual knots, Topology39 (2000) 1045–1068, 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). [ABSTRACT FROM AUTHOR]

In this paper, we construct a kind of new braided monoidal category over two Hom-Hopf algerbas (H , α) and (B , β) and associate it with two nonlinear equations. We first introduce the notion of an (H , B) -Hom-Long dimodule and show that the Hom-Long dimodule category H B L is an autonomous category. Second, we prove that the category H B L is a braided monoidal category if (H , α) is quasitriangular and (B , β) is coquasitriangular and get a solution of the quantum Yang-Baxter equation. Also, we show that the category H B L can be viewed as a subcategory of the Hom-Yetter-Drinfeld category H ⊗ B H ⊗ B H Y D . Finally, we obtain a solution of the Hom-Long equation from the Hom-Long dimodules. [ABSTRACT FROM AUTHOR]