Primality of low-crossing long virtual knots

Determine whether long virtual knots admitting diagrams with at most three classical crossings are prime.

Background

The paper applies its invariant to characterize long virtual knots that commute with all long virtual knots. For knots admitting diagrams with at most three classical crossings, the stated condition restricts the possibilities to the trivial knot K₀(1), K₃(13), and its vertical mirror K₃(13)#, all of which are classical.

The paper notes that every classical long virtual knot commutes with all long virtual knots. It explains that this conclusion could alternatively follow from Chrisman’s result on prime non-classical long virtual knots if all long virtual knots with at most three classical crossings were known to be prime. The authors explicitly leave the primality status of these low-crossing knots unresolved.