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 class of un-orthodox observables which breaks gauge invariance at
one point and which lead to a non-trivial theory on long knots in
$\mathbb{R}^3$. We have two main morals : 1. In the non-semi-simple case, there
is more to observe in Chern-Simons theory! There might 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. Our result is that the most general knot invariant
coming from pure BF topological quantum field theory is in the algebra
generated by the coefficients of the Alexander-Conway polynomial.