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 quantities8. In DNA knots, electrophoretic mobility appears to be correlated with the average crossing number of rope-length-minimizing configurations9, and a roughly linear empirical relation has been observed between the crossing number and rope length10. 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 number11. It can be shown that no smaller power is possible12,13,14.