ИСТИНА

About ten years ago, the first named author introduced the theory of free knots (previously conjectured by Turaev to be trivial) and found that this theory — a very rough simplification of the theory of virtual links — admitted new types of invariants never seen before: the invariants of links are valued in pictures, more precisely, in linear combinations of knot diagrams. For some classes of links, we have the formula [K] = K, where K in the LHS is our favourite link diagram (which is subject to various Reidemeister- like moves), and K in the RHS is the same diagram but seen as the rigid object. After some time it was noted that this approach does not work immediately for classical knots. In fact, the reason is that the approach when we look at “nodes” being “double” classical crossings is not the best one for classical knots. It is much better to look at “triple” crossings. In the present talk, we construct a map from equivalence classes of closed braids to 3-free knots and links (elder brothers of free knots and links). We also consider the map from links up to link-homotopy to 3-free links modulo some moves.