  The preservation of ambient isotopic equivalence under piecewise linear (PL)
approximation for smooth knots are prominent in molecular modeling and
simulation. Sufficient conditions are given regarding: (1) Hausdorff distance,
and (2) a sum of total curvature and derivative. High degree Bezier curves are
often used as smooth representations, where computational efficiency is a
practical concern. Subdivision can produce PL approximations for a given
B\'ezier curve, fulfilling the above two conditions. The primary contributions
are: (i) a priori bounds on the number of subdivision iterations sufficient to
achieve a PL approximation that is ambient isotopic to the original Bezier
curve, and (ii) improved iteration bounds over those previously established.
