  We analyse the possibility of defining complex valued Knot invariants
associated with infinite dimensional unitary representations of $SL(2,R)$ and
the Lorentz Group taking as starting point the Kontsevich Integral and the
notion of infinitesimal character. This yields a family of knot invariants
whose target space is the set of formal power series in $C$, which contained in
the Melvin-Morton expansion of the coloured Jones polynomial. We verify that
for some knots the series have zero radius of convergence and analyse the
construction of functions of which this series are asymptotic expansions by
means of Borel re-summation. Explicit calculations are done in the case of
torus knots which realise an analytic extension of the values of the coloured
Jones polynomial to complex spins. We present a partial answer in the general
case.
