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

TL;DR

This paper introduces an $O(n^2)$ algorithm to produce framed link presentations of 3-manifolds from special triangulations called solvable gems, enabling new computational approaches for manifold invariants.

Contribution

The paper presents a novel efficient algorithm linking triangulations to framed links, expanding computational tools for 3-manifold invariants and providing explicit presentations for known manifolds.

Findings

Algorithm operates in $O(n^2)$ time.

Framed link presentations obtained for several known 3-manifolds.

Potential to compute manifold invariants from triangulations.

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.