The aim of the program is to make invariants of topological objects "non-commutative", make
them valued "rather in 1-homotopy than homology" and to look at "characteristic cycles
instead of characteristic classes". In this talk I discussed only a part of that program:
just for usual knots, but potentially we can take whatever non simply connected manifold
(or moduli space) to study it by using these methods.
The key examples come from virtual knot spaces where one can easily calculate
invariants valued in "pictures" (diagrams of virtual knots) which are quite similar
to free groups (from the point of view of unique minimal representative).
These pictures contain much more powerful information than "polynomial", "homological"
or even "categorified" invariants.
Such structures naturally appear whenever the moduli spaces of studied objects
possess "nice real codimension 1-subspaces" which is often the case.