Random knotting in very long off-lattice self-avoiding polygons

TL;DR

The paper addresses knotting statistics in very long off-lattice self-avoiding polygons and tests a Poisson summand model for the number of prime knot factors, aiming to validate knot localization and the knot entropy conjecture. It combines scale-free pivot sampling with Clisby’s tree data structure and a fast knot-diagram simplification/classification pipeline (Knoodle) to generate and precisely classify extremely large SAPs, enabling measurements of $m_K^n$, $\lambda_K(n)$, and $R_K(n)$. The results show $P(m_K^n=m)$ is well described by a Poisson distribution with mean $\lambda_K(n)$ and that $R_K(n)=\lambda_K(n)/n$ follows finite-size corrections, yielding $N_{0_1}=656500.0 \pm 2500.0$ and universal amplitude ratios $C_K/C_{K'}$; most knotting consists of trefoil and figure-8 summands, supporting knot localization and the knot entropy conjecture. These findings corroborate previous on-lattice studies and establish a scalable framework for cross-model knotting analyses with potential extensions to non-SAP systems.

Abstract

We present experimental results on knotting in off-lattice self-avoiding polygons in the bead-chain model. Using Clisby's tree data structure and the scale-free pivot algorithm, for each $k$ between $10$ and $27$ we generated $2^{43-k}$ polygons of size $n=2^k$. Using a new knot diagram simplification and invariant-free knot classification code, we were able to determine the precise knot type of each polygon. The results show that the number of prime summands of knot type $K$ in a random $n$-gon is very well described by a Poisson distribution. We estimate the characteristic length of knotting as $656500 \pm 2500$. We use the count of summands for large $n$ to measure knotting rates and amplitude ratios of knot probabilities more accurately than previous experiments. Our calculations agree quite well with previous on-lattice computations, and support both knot localization and the knot entropy conjecture.

Figures (6)

• Figure 1: These plots show the probability $P(m_{3_1}(n)=m)$ of observing $m$ trefoil summands in off-lattice self-avoiding polygons of length $n$, together with the Poisson distribution function $\frac{(\lambda_K(n))^m e^{-\lambda_K(n)}}{m!}$ from \ref{['eq:fundamental equation']}. This is defined only at integer $m$, but we draw connecting curves as guides for the eye. The number of $n$-gons sampled ($2^{43}/n$) decreases with $n$, so the data appears rougher for large $n$.
• Figure 2: These plots show the probability $P(m_{4_1}(n)=m)$, $P(m_{5_1}(n)=m)$, $P(m_{5_2}(n)=m)$, and $P(m_{6_1}) = m)$ of observing $m$ summands of knot type $4_1$, $5_1/5_1^m$, $5_2/5_2^m$, or $6_1/6_1^m$ in off-lattice self-avoiding polygons of length $n$, together with the Poisson distribution functions $\frac{(\lambda_K(n))^m e^{-\lambda_K(n)}}{m!}$ from \ref{['eq:fundamental equation']}. Since these knots are much less probable than trefoils, only large values of $n$ are shown. The Poisson fit remains very good.
• Figure 3: This figure shows the total variation distance $\operatorname{TV}(p_1,p_2) = \sum_{m} |p_1(m) - p_2(m)|$ between the empirical distribution for the number of prime summands of knot type $K$ in an off-lattice self-avoiding $n$-gon $m_K^n$ and the corresponding Poisson model with mean $\lambda_K(n)$, using the estimate for $\lambda_K(n)$ from our dataset. We can see that almost all of these distances are very small, confirming that \ref{['eq:fundamental equation']} is an accurate approximation for these $K$ and $n$.
• Figure 4: The log-log plot at left shows the rate of knotting per edge: $R_K(n) = \lambda_K(n)/n$ estimated by our experiment. $99\%$ confidence intervals are included, but are too small to be visible on the plot. The fact that these values are becoming constant in $n$ supports our conclusion that the asymptotic knotting rate $C_K = \lim_{n \rightarrow \infty} \lambda_K(n)/n$ exists. Our estimates for $C_K$ appear with their knot types. The semilog plot at right shows this probability for trefoils only, along with our proposed fit to the finite-size correction \ref{['eq:rate with finite size effect model']}.
• Figure 5: Fraction of knots and unknots for $n$-edge SAPs, together with plots of $e^{-n/n_{0}}$ and $1-e^{-n/n_{0}}$ with the characteristic length $n_{0} = 656500.0$ estimated from knotting rates. The plot stops at $n=2^{23}$ because we observed no unknotted SAPs for larger $n$.