| We investigate the involutory Kuperberg invariant of oriented 3-manifolds and its algebraic and categorical frameworks. We revise the theory of diagrammatic morphisms in order to unify various approaches, justifying them via the theory of free structured categories. We interpret classical coherence theorems diagrammatically in terms of ordered incidence relations. We derive diagrammatic categories isomorphic to given structured categories. We interpret contraction diagrams as diagrammatic morphisms for strict symmetric categories that are possibly traced by means of a method for aligning contraction diagrams.;We revise and correct Kuperberg's diagrammatic theory of integrals for Hopf-algebra objects in traced, strict symmetric categories. In particular, we establish the equality of the left and right traces of (co)integrals, and the invariance of the (co)tracial (co)product under cyclic permutations of their entries, assuming the invertibility of the dimension scalar morphism and involutoryness. Hence, we eliminate redundancies in the original hypotheses on the ingredients for the involutory Kuperberg invariant.;We prove some basic properties of the involutory Kuperberg invariant (at the level of the universal involutory Hopf-algebra object of invertible dimension for traced, strict symmetric categories). We carefully examine and reformulate the reconstruction of 3-manifolds out of values of the invariant, also providing a counterexample to the augmentation in the original method. We make the reconstruction stricter in the case of values of the invariant at oriented, closed, connected 3-manifolds in order to recover manifolds of this kind instead of classes of bordisms. Then we reduce the Hopf-algebra axioms, upon reconstruction, to finite sequences of Singer moves, thus proving the completeness of the invariant for that class of 3-manifolds (not only for prime 3-manifolds, as initially proposed).;Finally, we develop methods for evaluating the involutory Kuperberg invariant from graph-encoded 3-manifolds and surgery instructions. We show that the invariant cannot detect sigma-symmetry of 3-manifolds systematically because that symmetry corresponds to algebraic properties of Hopf diagrams. We also establish steps towards the comparison of the Kuperberg and Kauffman-Radford-Hennings invariants, pointing out that the latter can be computed at the diagrammatic level, and providing a non-aligned, diagrammatic version of the Drinfel'd quantum double structure. |
