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

Abstract

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 $B\leftrightharpoons D$.

References in corpus (4)

• Twisted Link Theory
• Polynomial knot and link invariants from the virtual biquandle
• Roots in 3-manifold topology
• New Invariants of Long Virtual Knots

Cited by in corpus (2)

• On possible existence of HOMFLY polynomials for virtual knots
• Finite-type Invariants of Long and Framed Virtual Knots