  When a low dimensional chaotic attractor is embedded in a three dimensional
space its topological properties are embedding-dependent. We show that there
are just three topological properties that depend on the embedding: parity,
global torsion, and knot type. We discuss how they can change with the
embedding. Finally, we show that the mechanism that is responsible for creating
chaotic behavior is an invariant of all embeddings. These results apply only to
chaotic attractors of genus one, which covers the majority of cases in which
experimental data have been subjected to topological analysis. This means that
the conclusions drawn from previous analyses, for example that the mechanism
generating chaotic behavior is a Smale horseshoe mechanism, a reverse
horseshoe, a gateau roule, an S-template branched manifold, ..., are not
artifacts of the embedding chosen for the analysis.
