Motion Planning for Knot Untangling

Abstract

When given a very tangled but unknotted circular piece of string it is usually quite easy to move it around and tug on parts of it until it untangles. However, solving this problem by computer, either exactly or heuristically, is challenging. Various approaches have been tried, employing ideas from algebra, geometry, topology and optimization. This paper investigates the application of motion planning techniques to the untangling of mathematical knots. Such an approach brings together robotics and knotting at the intersection of these fields: rational manipulation of a physical model. In the past, simulated annealing and other energy minimization methods have been used to find knot untangling paths for physical models. Using a probabilistic planner, we have untangled some standard benchmarks described by over four hundred variables much more quickly than has been achieved with minimization. We also show how to produce candidates with minimal number of segments for a given knot. We discuss novel motion planning techniques that were used in our algorithm and some possible applications of our untangling planner in computational topology and in the study of DNA rings.