Reidemeister Moves and Groups
Recently, the author discovered an interesting class of knot-like objects called free knots. These purely combinatorial objects are equivalence classes of Gauss diagrams modulo Reidemeister moves (the same notion in the language of words was introduced by Turaev, who thought all free knots to be trivial). As it turned out, these new objects are highly non-trivial, and even admit non-trivial cobordism classes. An important issue is the existence of invariants where a diagram evaluates to itself which makes such objects "similar" to free groups: an element has its minimal representative which "lives inside" any representative equivalent to it.