Short version: Does anyone study equivalence classes generated by a given set of "moves" (in the sense of, but not limited to, Reidemeister moves) on the set of knot diagrams?
Yes, I understand that the concept of knot has a natural geometrical significance, that one usually views knot diagrams as a tool to study underlying knots, that the value of Reidemeister moves lies in how they preserve and generate the equivalence relation of isotopy. So, yes, I see why a knot theorist might reasonably have little interest, say, in looking at local moves not preserving isotopy.
So I'm asking this question in the spirit of abstraction for its own sake. But there is a precedent. A "symmetry theorist" studies groups because they capture the set of symmetries of important objects. But combinatorial group theory studies equivalence classes of strings under moves...and symmetry, if it enters the story at all, does so as a tool.
That said, it would be interesting if one could add an extra move to the Reidemeister moves that produced a coarser but computationally tractable classification.
No need to retread the ground here -- http://en.wikipedia.org/wiki/Reidemeister_move -- so, for example, I already understand that knot theorists know what happens with only Reidemeister type II and III moves. I am interested in stories like this, where one gets a finer equivalence relation that isotopy, but equally interested in sets of local moves that don't preserve isotopy and thus generate equivalence relations either coarser to, or simply incomparable with, isotopy.