On the algebraic structure of the set of long flat virtual knot diagrams with the connected sum operation

In [Y. H. Im, S. Kim and K. Lee, Embeddings from the set of long flat virtual knots to the set of long virtual knots and applications, J. Knot Theory Ramifications 29(5) (2020) 2050025], Im et al. introduced two embeddings from the set of long flat virtual knot diagrams to the set of long virtual knot diagrams so that we constructed invariants for long flat virtual knots via invariants for long virtual knots. We investigate one of the image sets of these embeddings in the set of long virtual knot diagrams and show examples that the connected sum operation is non-commutative in the set of long flat virtual knot diagrams.