We call the image of T N in R 3 an Nth degree trigonometric polynomial curve. We call the image of T N in this case a regular Nth degree trigonometric polynomial curve. An Nth degree harmonic knot is a knot with a regular Nth degree trigonometric polynomial parametrization. A knot K is tame if it is ambient isotopic to a simple, closed polygon in R 3. A knot is wild if it is not tame.
All harmonic knots are tame.
Definition 2.1 Let K be a tame knot. For all knots, the harmonic index is greater than or equal to 1 by definition. Let u be a fixed unit vector in R 3. The bridge index of the trefoil is 2. Let K be a tame knot.