All of the knots at the top of your post are already possible, because the Trefoil and the other ones mentioned are torus-knots:
The values of p and q can be specified independently, but there is of course a mathematical relationship between them when used to form the knot. So, in answer to your question, no, one is not a function of the other - they are both independent input parameters.
With regard to the SLICE parameters (not yet implemented), the idea was to create a cut of the knot between two angles, or more formally based on the "to be written" descriptive help:
Slice Start and End Angles
Allow the user to restrict the generation of the knot to only the portion that is contained within the volume of a right circular cylindrical sector with:
- Angle 𝛳, defined to be the absolute value of the difference between the angles specified and oriented accordingly
- Its axis made coincident with the axis of the virtual torus around which the knot is circumscribed
- An unbounded radius (or more specifically, a radius that is no smaller than the sum of the major and minor radii of the virtual torus [i.e., the distance from the central point of the torus along its axis to any of the equidistant points along its surface where the points lie at the farthest possible distance, radially from its center on the toric section formed by an imaginary equatorial plane used to bisect the torus]).
Or, in layman's terms, it allows for the creation of an arbitrary pizza-pie slice portion of the knot, just as it does for some of the built in parametric spline/polygonal shapes.
Also, thanks for the links to the Spiros plug-in. I can use that for ideas of what other features to add.