Shadows and quantum invariants

Abstract

This is a PhD thesis about low dimensional topology, in particular knot thory in 3-manifolds also different from the 3-sphere, topological applications of quantum invariants, and Turaev's shadows. There is an introduction and a survey for these topics. The thesis uses skein theory and focues on the connected sum of copies of S^1xS^2 and on the 3-tours as ambient manifolds. The skein space of the 3-torus is computed. The Tait conjecture and Eisermann's theorem have been extended to the connected sums of copies of S^1xS^2. The knots and links in S^1xS^2 whose crossing number is at most 3 are tabulated.