[math-fun] arc complexity of a knot

This discussion leads me to wonder that if a finite number of circular arcs in R^3 are fitted together to make a C^1 closed curve that's knotted, what is the smallest number of distinct arcs that can do this?

--Dan

--that might actually be an interesting new measure of knot complexity: the min number of arcs needed to make that knot. I can prove (but I'll instead leave it as a puzzle) ArcComplexity <= 2*PolygonalComplexity. Can a bound of the same nature be found (with some appropriate constant replacing "2") in the opposite direction?