Visualizing the tightening of knots

The study of physical models for knots has recently received much interest in the mathematics community. In this paper, we consider the ropelength model, which considers knots tied in an idealized rope. This model is interesting in pure mathematics, and has been applied to the study of a variety of problems in the natural sciences as well. Modeling and visualizing the tightening of knots in this idealized rope poses some interesting challenges in computer graphics. In particular, self-contact in a deformable rope model is a difficult problem which cannot be handled by standard techniques. In this paper, we describe a solution based on reformulating the contact problem and using constrained-gradient techniques from nonlinear optimization. The resulting animations reveal new properties of the tightening flow and provide new insights into the geometric structure of tight knots and links.