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.