Crossings and writhe of flexible and ideal knots

The data of ideal knots [Nature, 384, 142 (1996)] are reanalyzed and the average crossing number of the ideal knots $〈X{〉}_{\mathrm{ideal}}$ shows a nonlinear behavior with the essential crossing number C. Supplemented with our Monte Carlo simulations using the bond fluctuation model on flexible knotted polymers, our analysis indicates that $〈X{〉}_{\mathrm{ideal}}$ varies nonlinearly with both C and the corresponding average crossing number of the flexible knot, which is contrary to previous claims. Our extensive simulation data on the average crossing number of flexible knots suggest that it varies linearly with the square root of C. Furthermore, our data on the average writhe number $〈\mathrm{Wr}〉$ indicate that various knots are classified into holonomous groups, and $〈\mathrm{Wr}〉$ has a quantized linear increment with C in all four knot groups in our study.