Invariants in Quantum Geometry

In quantum geometry, we consider a set of loops, a compact orientable surface and a solid compact spatial region, all inside $$\mathbb{R}$ \times $\mathbb{R}$^3 \equiv $\mathbb{R}$^4$, which forms a triple. We want to define an ambient isotopic equivalence relation on such triples, so that we can obtain equivalence invariants. These invariants describe how these submanifolds are causally related to or `linked’ with each other, and they are closely associated with the linking number between links in ${\mathbb{R}}^3$. Because we distinguish the time-axis from spatial subspace in ${\mathbb{R}}^4$, we see that these equivalence relations, will also imply causality.