Computational topology for approximations of knots

Ji Li, T.J. Peters, K. E. Jordan

Abstract

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 Bézier 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 B\'ezier curve, and

       (ii) improved iteration bounds over those previously established.

 


Keywords

Knot approximation; ambient isotopy; Bézier curve; subdivision; piecewise linear approximation.

Subject classification

57Q37; 57M50; 41A15; 68R10.

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