Bibliographic record
Abstract
Virtual knot theory, introduced by Kauffman [Virtual Knot theory, European J. Combin.20 (1999) 663–690, arXiv:math.GT/9811028], is a generalization of classical knot theory of interest because its finite-type invariant theory is potentially a topological interpretation [D. Bar-Natan, u, v, w-knots: Topology, Combinatorics and low and high algebra] of Etingof and Kazhdan's theory of quantization of Lie bi-algebras [Quantization of Lie Bialgebras, I, Selecta Math. (N.S.) 2 (1996) 1–41, arXiv:q-alg/9506005]. Classical knots inject into virtual knots [G. Kuperberg, What is Virtual Link? Algebr. Geom. Topol.3 (2003) 587–591, arXiv:math.GT/0208039], and flat virtual knots [V. O. Manturov, On free knots, preprint (2009), arXiv:0901.2214; On free knots and links, preprint (2009), arXiv:0902.0127] is the quotient of virtual knots which equates the real positive and negative crossings, and in this sense is complementary to classical knot theory within virtual knot theory. We completely classify flat virtual tangles with no closed components (pure tangles). This classification can be used as an invariant on virtual pure tangles and virtual braids.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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".