Tight knot values deviate from linear relations

Applications of knots to the study of polymers have emphasized geometric measures on curves such as ‘energy’ 1 , 2 , 3 , 4 and ‘rope length’ 5 , 6 , 7 , which, when minimized over different configurations of a knot, give computable knot invariants related to physical quantities 8 . In DNA knots, electrophoretic mobility appears to be correlated with the average crossing number of rope-length-minimizing configurations 9 , and a roughly linear empirical relation has been observed between the crossing number and rope length 10 . Here we show that a linear relation cannot hold in general, and we construct infinite families of knots whose rope length grows as the 3/4 power of the crossing number 11 . It can be shown that no smaller power is possible 12 , 13 , 14 .