If knot theory was finitely presented, one could define knot invariants by assigning values to the generators so that the relations are satisfied. Well, knot theory is finitely presented, at least as a Vaughan Jones-style "planar algebra". We define a strange breed of planar algebras that can serve as the target space for an invariant defined along lines as above. Our objects appear to be simpler than the objects that appear in Drinfel'd theory of associators - our fundamental entity is the crossing rather than the re-association, our fundamental relation is the third Reidemeister move instead of the pentagon, and our "relations between relations" are simpler to digest than the Stasheff polyhedra. Yet our end product remains closely linked with Drinfel'd's theory of associators and possibly equivalent to it.
(joint with Dylan Thurston)
See the slides!