Combinatorial Dehn-Lickorish Twists and Framed Link Presentations of 3-Manifolds Revisited

Abstract

From a pseudo-triangulation with $n$ tetrahedra $T$ of an arbitrary closed orientable connected 3-manifold (for short, {\em a 3D-space}) $M^3$, we present a gem $J '$, inducing $\IS^3$, with the following characteristics: (a) its number of vertices is O(n); (b) it has a set of $p$ pairwise disjoint couples of vertices $\{u_i,v_i\}$, each named {\em a twistor}; (c) in the dual $(J ')^\star$ of $J '$ a twistor becomes a pair of tetrahedra with an opposite pair of edges in common, and it is named {\em a hinge}; (d) in any embedding of $(J ')^\star \subset \IS^3$, the $\epsilon$-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 $|G '|=M^3$; (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, {\em 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 $\IS^3$ and opens the way for an algorithmic method to actually obtaining the link by an $O(n^2)$-algorithm. This is the subject of a companion paper soon to be released.