  In the present paper, we address the problem how to get a map from knots in
the cylinder and on the thickened torus to some (generalisation of) virtual
knots called virtual-flat knots. The main construction takes a diagram on a
cylinder (torus) and adds some ``invisible'' crossings which gives rise to a
diagram which can be formally immersed but not embedded (drawn) on the cylinder
(torus) and living comfortably in thickened surfaces of higher genera. This
allows one to ``pull back'' invariants of virtual theory to the theory of knots
in the thickened cylinder (torus) where the parity bracket and other
picture-valued invariants are not strong enough.
