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It is well-known that the monoid of long virtual knots is not commutative.This contrasts with the case of classical long knots, where $A \# B\leftrightharpoons B \# A$ for all $A,B$. In the present paper, we present anew proof that two inequivalent non-classical prime long virtual knots nevercommute. The original result is due to Manturov. The techniques used here aremostly geometric. First, a slightly strengthened version of Kuperberg's theoremis established. We then show that a well-defined concatenation of two longknots in a thickened surface is preserved by stabilization when both long knotsare non-classical. Finally, it is proved that if $A,B,C,D$ are primenon-classical long virtual knots such that $A \# B$ is non-classical and $A \#B \leftrightharpoons C \# D$, then $A \leftrightharpoons C$ and $B\leftrightharpoons D$.

Prime Decomposition and Non-Commutativity in the Monoid of Long Virtual Knots