Evidence for Knot Localization in Very Long Self-Avoiding Polygons

Clayton Shonkwiler

Colorado State University

shonkwiler.org

/gs26

this talk!

AMS Special Session on Applications of Knot Theory

March 29, 2026

Joint Work With:

Jason Cantarella

University of Georgia

Funding

BIRS, Japan Science and Technology Agency (CREST JPMJCR19T4), National Science Foundation (DMS–2107700), Deutsche Forschungsgemeinschaft (320021702/GRK2326).

Tetsuo Deguchi

Ochanomizu University

Henrik Schumacher

RWTH Aachen University

Erica Uehara

Kyoto University

Take-Home Messages

We can determine the exact knot types of random self-avoiding polygons (SAPs) with 100M+ edges.
Counts of prime summands in random SAPs fit well to Poisson distributions.

Knot Localization

Localization Hypothesis

In the asymptotic limit, knots of type \(K\) in SAPs are independent events with a characteristic size and a fixed probability of starting at every edge of the polygon.

Two Conjectures

Summand Count Conjecture [Kesten/Hammersley 1960s(?)]

Suppose \(K\) is a prime knot type and let

\(m_K^n := \text{number of prime summands of knot type \(K\) in an \(n\)-gon}.\)

For \(n \gg 1\), \(m_K^n\) is approximately Poisson-distributed:

\(\mathbb{P}(m_K^n = m) \approx \frac{(\lambda_K(n))^m e^{-\lambda_K(n)}}{m!},\)

where \(\lambda_K(n)\) is the expected value of \(m_K^n\). Moreover, \(R_K(n) := \frac{\lambda_K(n)}{n} \to C_K\) as \(n \to \infty\) and, for \(K \neq K'\), \(m_K^n\) and \(m_{K'}^n\) are approximately independent.

Knot Entropy Conjecture [Orlandini, Tesi, Janse van Rensburg, Whittington 1996]

For any knot type \(K\) and \(n \gg 1\),

\(\mathbb{P}_{0_1}(n) \approx e^{-n/n_0} \quad \text{and} \quad \mathbb{P}_K(n) \approx C_K n^{m(K)}e^{-n/n_0} \left(1 + \frac{\beta_K}{n^\Delta} + \frac{\gamma_K}{n}\right)\).

\(n_0\) is called the characteristic length of knotting.

Note: KEC has been proved for lattice polygons in tubes by Beaton et al. (2024).

SCC Implies KEC

\(\mathbb{P}_{0_1}(n)=\mathbb{P}(m_K^n=0 \text{ for all prime }K)\)

\(\approx \Pi_K \mathbb{P}(m_K^n = 0)\)

\(= \Pi_{K} e^{-R_K(n) n}\)

\(= e^{-\left(\sum_K R_K(n)\right) n}\)

Let \(R_K(n) = \lambda_K(n)/n\) be the rate of knot production, so SCC says

\(\mathbb{P}(m_K^n = m) \approx \frac{(R_K(n)n)^m e^{-R_K(n)n}}{m!}.\)

Unknot Probability

\(= e^{-n/n_0}\)

with \(n_0 = \frac{1}{\sum_K R_K(n)}\)

Prime Knot Probability

\(\mathbb{P}_{K}(n)=\mathbb{P}(m_K^n = 1 \text{ and }m_{K'}^n=0 \text{ for }K' \neq K)\)

\(\approx\mathbb{P}(m_K^n=1) \times \Pi_{K'} \mathbb{P}(m_{K'}^n = 0)\)

\(\approx R_K(n)n e^{-R_K(n)n}\Pi_{K'} e^{-R_{K'}(n) n}\)

\(= R_K(n)n e^{-\left(\sum_{K'} R_{K'}(n)\right) n}\)

\(= R_K(n)n e^{-n/n_0}\)

A Plan for Testing SCC

Generate SAPs for \(n = 2^{10}=1024\) to \(2^{27}=134,\!217,\!728\).
Count the number of summands of each prime knot type.
See if the empirical distribution is approximately Poisson.
If so, see if the rates \(R_K(n)\) limit to constants.
Use \(R_K(n)\) rather than \(\mathbb{P}_K(n)\) to measure \(C_K\) in the large-\(n\) limit.