Clayton Shonkwiler PRO
Mathematician and artist
Evidence for Knot Localization in Very Long Self-Avoiding Polygons
/gs26
this talk!
AMS Special Session on Applications of Knot Theory
March 29, 2026
Joint Work With:
University of Georgia
Funding
BIRS, Japan Science and Technology Agency (CREST JPMJCR19T4), National Science Foundation (DMS–2107700), Deutsche Forschungsgemeinschaft (320021702/GRK2326).
Ochanomizu University
RWTH Aachen University
Kyoto University
Take-Home Messages
- We can determine the exact knot types of random self-avoiding polygons (SAPs) with 100 million 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:
\(P(m_K^n = m) \approx \frac{(\lambda_K(n))^m e^{-\lambda_K(n)}}{m!},\)
where \(\lambda_K(n)\) is the expected value of \(m_K^n\).
Knot Entropy Conjecture [Orlandini, Tesi, Janse van Rensburg, Whittington 1996]
For any knot type \(K\) and \(n \gg 1\),
\(P_{0_1}(n) \approx e^{-n/n_0} \quad \text{and} \quad P_K(n) \approx C_K n^{m(K)}e^{-n/n_0} \left(1 + \frac{\beta_K}{n^\Delta} + \frac{\gamma_K}{n}\right)\).
\(n_0\) 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
\(P_{0_1}(n)\)
\(\approx \Pi_K P(m_K^n = 0)\)
\(= \Pi_{K} e^{-R_K(n) n}\)
\(= e^{-\left(\sum_K R_K(n)\right) n}\)
Let \(R_K(n) = \lambda_K(n)/n\) be the rate of knot production, so SCC says
\(P(m_K^n = m) \approx \frac{(R_K(n)n)^m e^{-R_K(n)n}}{m!}.\)
Unknot Probability
\(= e^{-n/n_0}\)
with \(n_0 = \frac{1}{\sum_K R_K(n)}\)
Prime Knot Probability
Thank you!
shonkwiler.org/gs26
References
Random knotting in very long off-lattice self-avoiding polygons
Jason Cantarella, Tetsuo Deguchi, Henrik Schumacher, Clayton Shonkwiler, and Erica Uehara
Hard unknots are often easy from a different perspective
Jason Cantarella, Henrik Schumacher, and Clayton Shonkwiler
In preparation
Evidence for Knot Localization in Very Long Self-Avoiding Polygons
By Clayton Shonkwiler
