Shadows and quantum invariants

We investigate quantum invariants and their topolological applications through skein theory and the use of Turaev's shadows. We study knots and links in 3-manifold different from S^3, in particular we focus on the connected sum #_g(S^1xS^2) of g>=0 copies of S^1xS^2 and on the 3-torus T^3. Our main tools are the Kauffman bracket an the Turaev's shadows. An introductin and a surey to skein theory and Turaev's shadows is given. We present all the main open conjectures about topological applications of quantum invariants. Two theorems about links in S^3 are extended to links and colored knotted trivalent graphs in #_g(S^1xS^2). The first one is the Tait conjecture about crossing number and alternating links, and the second one is the Eisermann's theorem about ribon surfaces. Both are topological applications of the Jones polynomial. We compute the skein space of the 3-torus. Moreover we show the table of knots a links in S^1xSì2 with crossing number up to 3.