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

Abstract

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.