Embedding of 2-dimensional cell complexes in \(S^ 3\) determined by their 1-skeletons (Q1819791)

We prove the following theorem: A polyhedral embedding of a 2-dimensional cell complex in \(S^ 3\) is determined up to ambient isotopy rel the 1- skeleton by the embedding of the 1-skeleton, provided the cell complex is ''proper'' and ''fine enough''. Applications of the theorem are given in distinguishing certain graphs in \(S^ 3\) from their mirror images. (This is of interest to chemists studying stereoisomerism.) Examples are given to illustrate that the theorem can fail without either hypothesis ''proper'' or ''fine enough''. The main theorem may be generalized by replacing \(S^ 3\) by an irreducible 3-manifold with nonempty boundary.