Integral face-edge duality for knot triangulations

For an ordered ideal triangulation of a knot exterior in $S^3$, we give a cochain construction relating its face matrix $C$ to its preferred-longitude gluing matrices $A,B$. The obstruction to uniqueness is the first cohomology of the ideal compactification. Its vanishing yields nullity doubling and the FAMED matrix identity, including a constrained formulation for singular face matrices. A separate rank argument under a strict angle structure supplies a proposed proof of Wong’s generalized FAMED conjecture. Over the integers, the construction gives a direct-sum decomposition of ${\ker }_{\mathbb{Z}}B$ and an exact sequence from the two face cokernels to the gluing cokernel, with quotient $\mathbb{Z}\mathrm{/}2\mathbb{Z}$. In the unhalved longitude convention this implies $\mathrm{\mid }\det B\mathrm{\mid }=2\mathrm{\mid }\det C{\mathrm{\mid }}^2$. We also obtain cyclic face torsion, order-reversal invariance of Smith normal form, and affine face coordinates for fixed lifted gluing systems. The determinant-one assertion and the full Andersen–Kashaev volume conjecture remain open here.