PROPERTIES OF IDEAL COMPOSITE KNOTS

The shortest tube of constant diameter that can form a given knot repr esents the 'ideal' form of the knot(1,2). Ideal knots provide an irred ucible representation of the knot, and they have some intriguing mathe matical and physical features, including a direct correspondence with the time-averaged shapes of knotted DNA molecules in solution(1,2). He re we describe the properties of ideal forms of composite knots-knots obtained by the sequential tying of two or more independent knots (cal led factor knots) on the same string. We find that the writhe (related to the handedness of crossing points) of composite knots is the sum o f that of the ideal forms of the factor knots. By comparing ideal comp osite knots with simulated configurations of knotted, thermally fluctu ating DNA, we conclude that the additivity of writhe applies also to r andomly distorted configurations of composite knots and their correspo nding factor knots. We show that composite knots with several factor k nots may possess distinct structural isomers that can be interconverte d only by loosening the knot.