  For applications in computing, Bezier curves are pervasive and are defined by
a piecewise linear curve L which is embedded in R^3 and yields a smooth
polynomial curve C embedded in R^3. It is of interest to understand when L and
C have the same embeddings. One class of counterexamples is shown for L being
unknotted, while C is knotted. Another class of counterexamples is created
where L is equilateral and simple, while C is self-intersecting. These
counterexamples were discovered using curve visualizing software and numerical
algorithms that produce general procedures to create more examples.
