Reidermeister's theorem states that the set of knots, modulo ambient isotopy, is isomorphic to the planar algebra generated by crossings, modulo Reidemeister moves. This planar algebra presentation is the starting point for much of quantum topology. Of course this set of generators and relations isn't unique. I'm interested in unknotting moves other than crossing changes, and I would like to ask

Is there another known "convenient" planar algebra presentation, generators modulo relations, which gives rise to knots? In particular, can I sensibly choose generators corresponding to resolutions of triple-points?

We can generalize in many ways. For example we can allow circuit algebras, which are non-planar, and obtain the set of virtual knots. I have the same question regarding such generalizations. Also

Is there a result that any presentation of a planar algebra giving rise to knots, other than the one given by crossings modulo Reidemeister moves, would necessarily be significantly harder to work with? I.e. is there some sort of non-trivial "optimality result" for the presentation "crossings mod Reidemeister moves"?