  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 a
new proof that two inequivalent non-classical prime long virtual knots never
commute. The original result is due to Manturov. The techniques used here are
mostly geometric. First, a slightly strengthened version of Kuperberg's theorem
is established. We then show that a well-defined concatenation of two long
knots in a thickened surface is preserved by stabilization when both long knots
are non-classical. Finally, it is proved that if $A,B,C,D$ are prime
non-classical long virtual knots such that $A \# B$ is non-classical and $A \#
B \leftrightharpoons C \# D$, then $A \leftrightharpoons C$ and $B
\leftrightharpoons D$.
