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 machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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 source (direct Gemma or distilled Codex), 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".