Quantum invariants based on ideal triangulations (Intelligence of Low-dimensional Topology)

Abstract

We have presented three types of reconstructions of the universal invariant. The original construction associates the universal R-matrix or its inverse with each crossing of link diagrams, while the reconstructions associate the S-tensor or its inverse with each ideal tetrahedron of 3-manifolds. The first reconstruction (Theorem 3.1) uses slice diagrams of tangles and provides a topological realization of Kashaev’s embedding at each crossing of tangles. However, it cannot be extended to be an invariant of 3-manifolds. By introducing integral normal o-graphs, we can represent framed 3-manifolds. In particular, for closed framed 3-manifolds with a vanishing first Betti number, we establish a one-to-one correspondence (Theorem 4.1). This allows us to construct an invariant of closed framed 3-manifolds (Theorem 5.3). Taking the framing structure into account, in the context of the universal invariant, we provide two alternative reconstructions (Theorems 6.2 and 6.3) using integral normal o-graphs. This means that the invariant Z extends the universal invariant in a three dimensional manner. We anticipate that our framework will provide a new approach to studying quantum invariants in a three-dimensional context.