Let be given a { \em colored 3-pseudo-triangulation} $\mathcal{ H}^\star$ with $n$ tetrahedra. Colored means that each tetrahedron have vertices distinctively colored 0,1,2,3. In a { \em pseudo} 3-triangulation the intersection of simplices might be subsets of simplices of smaller dimensions (faces), instead of a single maximal face, as for true triangulations. If $\mathcal{ H}^\star$ is the dual of a cell 3-complex induced (in an specific way to be made clear) by a pair of Jordan curves with $2n$ transversal crossings, then we show that the induced 3-manifold $|\mathcal{ H}^\star|$ is $\mathbb{ S}^3$ and we make available an $O(n^2$)-algorithm to produce a PL-embedding (\cite{ rourke1982introduction}) of $\mathcal{ H}^\star$ into $\mathbb{ S}^3$. This bound is rather surprising because such PL-embeddings are often of exponential size. This work is the first step towards obtaining, via an $O(n^ 2)$-algorithm, a framed link presentation inducing the same closed orientable 3-manifold as the one given by a colored pseudo-triangulation. Previous work on this topic appear in \url{ http://arxiv.org/abs/1212.0827}, \cite{ linsmachadoA2012}, \url{ http://arxiv.org/abs/1212.0826}, \cite{ linsmachadoB2012} and \url{ http://arxiv.org/abs/1211.1953}, \cite{ linsmachadoC2012}. However, the exposition and the new proofs of this paper are meant to be entirely self-contained.
Categories: math.GT