Distinguishing Physical and Mathematical Knots and Links

Distinguishing Physical and Mathematical Knots and Links Joel Hass (joint work with Alexander Coward ) The theory of knots and links studies one-dimensional submanifolds of R. These are often described as loops of string, or rope, with their ends glued together. Real ropes however are not one-dimensional, but have a positive thickness and a finite length. Indeed, most physical applications of knot theory are related more closely to the theory of knots of fixed thickness and length than to classical knot theory. For example, biologists are interested in curves of fixed thickness and length as a model for DNA and protein molecules. In these applications the thickness of the curve modeling the molecule plays an essential role in determining the possible configurations. The two most fundamental problems concerning physical knots and links are to show the existence of a Gordian Unknot and a Gordian Split Link. A Gordian Unknot is a loop of fixed thickness and length whose core is unknotted, but which cannot be deformed to a round circle by an isotopy fixing its length and thickness. A Gordian Split Link is a pair of loops of fixed thickness whose core curves can be split, or isotoped so that its two components are separated by a plane, but cannot be split by an isotopy fixing each component’s length and thickness. In recent joint work with Alexander Coward, we established the existence of a Gordian Split Link. We thus showed for the first time that the theory of physically realistic curves of fixed thickness and length is distinct from the classical theory of knots and links. Theorem 1. A Gordian Split Link exists. Figure 1. A Gordian Split Link.