Shadows and quantum invariants

TL;DR

This thesis explores low-dimensional topology, focusing on quantum invariants, skein theory, and shadows, with new results on skein spaces, conjectures, and knot tabulations in specific 3-manifolds.

Contribution

It extends the Tait conjecture and Eisermann's theorem to connected sums of S^1xS^2 and computes skein spaces for the 3-torus, providing new insights into knots in these manifolds.

Findings

Skein space of the 3-torus computed

Tait conjecture extended to connected sums of S^1xS^2

Knots with crossing number ≤ 3 in S^1xS^2 tabulated

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.