Bibliographic record
Abstract
We construct a family of trivial [Formula: see text]-knots [Formula: see text] in [Formula: see text] such that the maximal complexity of [Formula: see text]-knots in any isotopy connecting [Formula: see text] with the standard unknot grows faster than a tower of exponentials of any fixed height of the complexity of [Formula: see text]. Here, we can either construct [Formula: see text] as smooth embeddings and measure their complexity as the ropelength (a.k.a the crumpledness) or construct PL-knots [Formula: see text], consider isotopies through PL knots, and measure the complexity of a PL-knot as the minimal number of flat [Formula: see text]-simplices in its triangulation. These results contrast with the situation of classical knots in [Formula: see text], where every unknot can be untied through knots of complexity that is only polynomially higher than the complexity of the initial knot.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 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) | 0.001 | 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".