Parity and cobordism of free knots

A simple invariant is constructed which obstructs a free knot to be truncated. In particular, this invariant provides an obstruction to the truncatedness of curves immersed in two-dimensional surfaces. A curve on an oriented two-dimensional surface S-g is referred to as truncated (null-cobordant) if there exists a three-dimensional manifold M with boundary S-g and a smooth proper map of a two-disc to M such that the image of the boundary of the disc coincides with the curve. The problem of truncatedness for free knots is solved in this paper using the notion of parity recently introduced by the author.