Virtual knot theory methods in classical topology. Part 2: Classical links on the cylinder

This talk is a part of the project of creating “non-commutative” invariants in topology. The main idea is to replace “characteristic classes” of moduli spaces with “characteristic loops”. We discuss “the last stage” of the talk devoted to the abstract objects we get in the end: the free knots, and discuss their invariants valued in free groups. These invariants allow one to detect easily mutations, invertibility, and other phenomena. We shall use parity on the cylinder to construct classical link invariants valued in free groups.