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

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

Framed link presentations of 3-manifolds by an O(n2)O(n^2) algorithm, I: gems and their duals

Given an special type of triangulation $T$ for an oriented closed 3-manifold $M^3$ we produce a framed link in $S^3$ which induces the same $M^3$ by an algorithm of complexity $O(n^2)$ where $n$ is the number of tetrahedra in $T$ . The special class is formed by the duals of the {\em solvable gems}. These are in practice computationaly easy to obtain from any triangulation for $M^3$. The conjecture that each closed oriented 3-manifold is induced by a solvable gem has been verified in an exhaustible way for manifolds induced by gems with few vertices. Our algorithm produces framed link presentations for well known 3-manifolds which hitherto did not one explicitly known. A consequence of this work is that the 3-manifold invariants which are presently only computed from surgery presentations (like the Witten-Reshetkhin-Turaev invariant) become computable also from triangulations. This seems to be a new and useful result. Our exposition is partitioned into 3 articles. This first article provides our motivation, some history on presentation of 3-manifolds and recall facts about gems which we need.Comment: This is a minor revision of part 1 with 11 pages and 7 figures of a 3-part articl