Abstract:We study the Chern-Simons topological quantum field theory with an inhomogeneous gauge group, a non-semi-simple group obtained from a semi-simple one by taking its semi-direct product with its Lie algebra. We find that the standard knot observables (i.e. traces of holonomies along knots) essentially vanish, but yet, the non-semi-simplicity of our gauge group allows us to consider a new class of un-orthodox observables which breaks gauge invariance at one point and which lead to a non-trivial theory. We have two main morals : 1. In the non-semi-simple case, there's a little more to observe in Chern-Simons theory! There may be other interesting non semi-simple gauge groups to study in this context beyond our example. 2. In our case of an inhomogeneous gauge group, we find that Chern-Simons theory with the un-orthodox observable is actually the same as 3D BF theory with the Cattaneo-Cotta-Ramusino-Martellini knot observable. This leads to a simplification of their results and enables us to generalize and solve a problem they posed regarding the relation between BF theory and the Alexander-Conway polynomial.