From a pseudo-triangulation with n tetrahedra T of an arbitrary closed orientable connected 3-manifold (for short, a 3D-space) M3, we present a gem J′, inducing 𝕊3, with the following characteristics: (a) its number of vertices is O(n); (b) it has a set of p pairwise disjoint couples of vertices {ui, vi}, each named a twistor; (c) in the dual (J′)⋆ of J′ a twistor becomes a pair of tetrahedra with an opposite pair of edges in common, and it is named a hinge; (d) in any embedding of (J′)⋆ ⊂ 𝕊3, the ∊-neighborhood of each hinge is a solid torus; (e) these p solid tori are pairwise disjoint; (f) each twistor contains the precise description on how to perform a specific surgery based in a Denh–Lickorish twist on the solid torus corresponding to it; (g) performing all these p surgeries (at the level of the dual gems) we produce a gem G′ with |G′| = M3; (h) in G′ each such surgery is accomplished by the interchange of a pair of neighbors in each pair of vertices: in particular, |V(G′) = |V(J′)|.
This is a new proof, based on a linear polynomial algorithm, of the classical theorem of Wallace (1960) and Lickorish (1962) that every 3D-space has a framed link presentation in 𝕊3 and opens the way for an algorithmic method to actually obtaining the link by an O(n2)-algorithm. Actually this has been done and awaits a proper implentation.