The 3D index of an ideal triangulation and angle structures

The 3D index of Dimofte–Gaiotto–Gukov is a partially defined function on the set of ideal triangulations of 3-manifolds with r tori boundary components. For a fixed 2r tuple of integers, the index takes values in the set of q-series with integer coefficients. Our goal is to give an axiomatic definition of the tetrahedron index and a proof that the domain of the 3D index consists precisely of the set of ideal triangulations that support an index structure. The latter is a generalization of a strict angle structure. We also prove that the 3D index is invariant under 3–2 moves, but not in general under 2–3 moves.