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

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 $B \leftrightharpoons D$.