ABSTRACT
The two main arguments that can be taken into account in the theory of "long" virtual knots and could not be used before are the following: One can indicate the initial and the final arcs of the quandle; the elements corresponding to them are invariant under generalised Reidemeister moves. One can take two different quandle-like structures of the same type at vertices depending on which arc is “before” and which are "after" according to the orientation of a long knot. The chapter aims to construct invariants of long virtual knots that feel "the breaking point". The proof is quite analogous to the invariance proof of the virtual quandle. Thus, the details will be sketched. The invariance under purely virtual moves and the semi virtual move goes as in the classical case.
