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)