Representations of non-singular planar tangles by operators

In this paper we study how to distinguish two embeddings of a finite collection of disjoint circles into the plane up to planar isotopy. We adopt the spirit of the approach by V. Turaev, Operator Invariants of Tangles, Math. USSR-Izv. 35 (1990), 411--444, by considering a category of planar tangles and representing it in an ``algebraic'' category. From this we can extract a numerical invariant for embeddings of a finite collection of disjoint circles and this invariant is, up to certain choices, complete.