We describe a method that determines directly the topological configuration of the intersection curve $\mathcal{C}$ between two parametric surfaces, represented in the 4D parameter space and given by 3 equations $F_i(s,t) = G_i(u,v)$, $i = 1, 2, 3$ with variables in $[0..1]4$.

Our algorithm certifies the exactness of the topology of the curve up to any specified precision. It relies on efficient solvers of systems of polynomial equations and generalizes similar studies obtained for curves in 2D and 3D, relying on subdivision and exclusion techniques.