Relative knot probabilities in confined lattice polygons
Bibliographic record
Abstract
In this paper we examine the relative knotting probabilities in a lattice model of ring polymers confined in a cavity. The model is of a lattice knot of size n in the cubic lattice, confined to a cube of side length L and with volume V=(L+1)^{3} sites. We use Monte Carlo algorithms to estimate approximately the number of conformations of lattice knots in the confining cube. If p_{n,L}(K) is the number of conformations of a lattice polygon of length n and knot type K in a cube of volume L^{3}, then the relative knotting probability of a lattice polygon to have knot type K, relative to the probability that the polygon is the unknot (the trivial knot, denoted by 0_{1}), is ρ_{n,L}(K/0_{1})=p_{n,L}(K)/p_{n,L}(0_{1}). We determine ρ_{n,L}(K/0_{1}) for various knot types K up to six crossing knots. Our data show that these relative knotting probabilities are small over a wide range of the concentration φ=n/V of monomers for values of L≤12 so that the model is dominated by unknotted lattice polygons. Moreover, the relative knot probability increases with φ along a curve that flattens as the Hamiltonian state is approached.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".