Unknots with highly knotted control polygons

An example is presented of a cubic Bézier curve that is the unknot (a knot with no crossings), but whose control polygon is knotted. It is also shown that there is no upper bound on the number of crossings in the control polygon for an unknotted Bézier curve. These examples complement known upper bounds on the number of subdivisions sufficient for a control polygon to be ambient isotopic to its Bézier curve. There can be substantial topological differences between a curve and its control polygon, as depicted in Figure 1, which has control polygon P0, P1, . . . P5, P0.