  Spin Foam and Loop approaches to Quantum Gravity reformulate Einstein's
theory of relativity in terms of connection variables. The metric properties
are encoded in face bivectors/conjugate fluxes that are required to satisfy
certain conditions, in order to allow for their geometric interpretation. We
show that the (sub-)set of the so-called `volume simplicity constraints' is not
implemented properly in the current EPRL-FK spinfoam vertex amplitude, if
extended to arbitrary polyhedra. We then propose that a certain knot-invariant
of the bivector geometry, induced on the boundary graph, encodes the missing
conditions, allowing for reconstruction of a polytope from its two-dimensional
faces. Implemented in the quantum amplitude, this leads to corrected
semi-classical asymptotics for a hypercuboid, and is conjectured to be
non-trivial in more general situations. The analysis of linear version of
`volume simplicity' suggests to switch from hypersurface normals to edge
lengths, that is from 3-forms directly to tetrads -- in the extended
configuration space of the Plebanski constrained formulation. We then give the
corresponding dual version of linear simplicity constraints, which prescribe 3d
volume for the polyhedral faces in the boundary of a 4d polytope. We also
analyse the status of metric/vielbein degrees of freedom and the role of local
translations in the classical Einstein-Cartan gravity, viewed as a Poincare
gauge theory. The relation with the diffeomorphism symmetry is established
through the key concept of development, which generalizes parallel transport of
vectors in the geometric theory of Cartan connections. We advocate the latter
to be the natural gauge-theoretic framework for the theory of relativity.
