Ranking Knots of Random, Globular Polymer Rings

An analysis of extensive simulations of interacting self-avoiding polygons on cubic lattice shows that the frequencies of different knots realized in a random, collapsed polymer ring decrease as a negative power of the ranking order, and suggests that the total number of different knots realized grows exponentially with the chain length. Relative frequencies of specific knots converge to definite values because the free energy per monomer, and its leading finite size corrections, do not depend on the ring topology, while a subleading correction only depends on the crossing number of the knots.

Figure 1

Probability of finding an unknot. The straight dashed line represents the exponential fit with $N_0=420$.

Figure 2

Probability of rank-ordered knots for $N=600$. Curves are for sample sizes increasing as a power of 2, from ${10}^4$ to $64\times {10}^4$. Some simple knot topologies (unknot, $3_1$, $4_1$, $5_2$, and $5_1$) and a more complicated one ($6_1\#{10}_{128}$) are also associated with their ranking.

Figure 3

(a) Power-law ranges of $P_q(N)/P_{\varnothing }(N)$, for $N=200$, 400, 500, 600, 700, 800, 1000, 1200, 1400, 1600, and 1800. For $N=1800$ also, the cutoff, due to limited sampling, is shown. (b) Extrapolation of the exponent of the Zipf law for $N\to \infty$.

Figure 4

Probability of a knot type $k$ over the probability of an unknot, vs $1/N$, for some knots. Each ratio converges towards an asymptotic value $A_k/A_{\varnothing }$. Straight lines are fits $\sim \exp [-({\delta }_k-{\delta }_{\varnothing })/N]$.

Figure 5

Fitted ${\delta }_k-{\delta }_{\varnothing }$ as a function of the minimal crossing number $n_c$ for knots $3_1$ ($n_c=3$), $4_1$ ($n_c=4$), $5_1$ and $5_2$ ($n_c=5$), $3_1\#3_1$, $6_1$, $6_2$, and $6_3$ ($n_c=6$), and for all knots with $n_c=7$. (Large circles denote averages for each $n_c$.) ${\delta }_k$’s are grouped in bands corresponding to $n_c$’s, i.e., to a good approximation, they depend only on the number of essential crossings of a knot type. A fit is also shown, suggesting that ${\delta }_k-{\delta }_{\varnothing }$ grows as a power of $n_c$.