Grasping and folding knots

String stretched tightly along a sequence of fixed grasp points takes the shape of a polygonal arc. In this work, we investigate how many points are necessary and sufficient to grasp and tie arbitrary knots while maintaining tension, so that the string remains polygonal. This approach allows reasoning that is entirely geometric, which does not rely on potentially inaccurate dynamic models of the string or detailed knowledge of physical characteristics of the string. Algorithms are proposed to determine the contact locations, and generate the motions needed to tie arbitrary knots. This work shows that a number of grasp points that is linear in the number of crossings in a knot diagram is sufficient to immobilize string in a polygonal shape with the topology of an arbitrary knot, or to fold or unfold the knot from a straight configuration.