Invariants of tangles

A tangle is a pair of strings ( t 0 , t 1 ) properly embedded in a 3-ball. Tangles have been used in several approaches to the classification of knots (see [ 1, 4, 15 ]). In these investigations, one keeps track of the endpoints of the arcs, so that the sum of two tangles along their boundaries is well defined. In particular, the sum of a given tangle with a trivial tangle, and any invariants of the resulting link, are invariants of the tangle under the restricted relation of isotopy keeping the endpoints fixed.