Quantum topology and the Lorentz group - Martins, João Nuno Gonçalves Faria (2004)
We analyse the perturbative expansion of knot invariants related with infinite dimensional representations of sl(2,R) and the Lorentz group taking as a starting point the Kontsevich Integral and the notion of central characters of infinite dimensional unitary representations of Lie Groups. The prime aim is to define C-valued knot invariants. This yields a family of C([h]]-valued knot invariants contained in the Melvin-Morton expansion of the Coloured Jones Polynomial. It is verified that for some knots, namely torus knots, the power series obtained have a zero radius of convergence, and therefore we analyse the possibility of obtaining analytic functions of which these power series are asymptotic expansions by means of Borel re-summation. This process is complete for torus knots, and a partial answer is presented in the general case, which gives an upper bound on the growth of the coefficients of the Melvin-Morton expansion of the Coloured Jones Polynomial. In the Lorentz group case, this perturbative approach is proved to coincide with the algebraic and combinatorial approach for knot invariants defined out of the formal R-matrix and formal ribbon elements in the Quantum Lorentz Group, and its infinite dimensional unitary representations.