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Let be given a {\em colored 3-pseudo-triangulation} $\mathcal{H}^\star$ with$n$ tetrahedra. Colored means that each tetrahedron have vertices distinctivelycolored 0,1,2,3. In a {\em pseudo} 3-triangulation the intersection ofsimplices might be subsets of simplices of smaller dimensions (faces), insteadof a single maximal face, as for true triangulations. If $\mathcal{H}^\star$ isthe dual of a cell 3-complex induced (in an specific way to be made clear) by apair of Jordan curves with $2n$ transversal crossings, then we show that theinduced 3-manifold $|\mathcal{H}^\star|$ is $\mathbb{S}^3$ and we makeavailable 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 ofexponential size. This work is the first step towards obtaining, via an $O(n^2)$-algorithm, a framed link presentation inducing the same closed orientable3-manifold as the one given by a colored pseudo-triangulation. Previous work onthis 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 thispaper are meant to be entirely self-contained.

PL-embedding the dual of two Jordan curves into $\mathbb{S}^ 3$ by an $O(n^ 2)$-algorithm