Four-thirds power law for knots and links

Physical knot theory has recently been applied to polymer dynamics, and specifically to gel electrophoresis of DNA1,2. Knot energies3,4,5,6 measure the complexity of a knot conformation; minimum energy conformations are considered canonical or ‘ideal’ conformations. The rope length of a knot is one such measure of energy6, and an approximately linear relationship between rope length and the average crossing number for minimum rope-length conformations of simple knots has been reported7. Here I show that a linear relationship cannot hold in general: the rope length required to tie an N-crossing knot or link varies at least between ˜N3/4 and ˜N.