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

TL;DR

This paper presents an $O(n^2)$-algorithm for PL-embedding a dual pseudo-triangulation derived from two Jordan curves into the 3-sphere, enabling efficient visualization of certain 3-manifolds.

Contribution

It introduces a quadratic-time algorithm for embedding dual pseudo-triangulations into $ ext{S}^3$, a significant improvement over the typically exponential size of such embeddings.

Findings

The induced 3-manifold is $ ext{S}^3$.

The algorithm runs in $O(n^2)$ time.

Provides a foundation for framed link presentations of 3-manifolds.

Abstract

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.