  We generalize Kauffman's famous formula defining the Jones polynomial of an
oriented link in 3-space from his bracket and the writhe of an oriented
diagram. Our generalization is an epimorphism between skein modules of tangles
in compact connected oriented 3-manifolds with markings in the boundary.
Besides the usual Jones polynomial of oriented tangles we will consider graded
quotients of the bracket skein module and Przytycki's q-analogue of the first
homology group of a 3-manifold. In certain cases, for example for links in
submanifolds of rational homology 3-spheres, we will be able to define an
epimorphism from the Jones module onto the Kauffman bracket module. For the
general case we define suitably graded quotients of the bracket module, which
are graded by homology. The kernels define new skein modules measuring the
difference between Jones and bracket skein modules. We also discuss gluing in
this setting.
