  We construct a family of rings. To a plane diagram of a tangle we associate a
complex of bimodules over these rings. Chain homotopy equivalence class of this
complex is an invariant of the tangle. On the level of Grothendieck groups this
invariant descends to the Kauffman bracket of the tangle. When the tangle is a
link, the invariant specializes to the bigraded cohomology theory introduced in
our earlier work.
