Bibliographic record
Abstract
There is a well-known injective homomorphism [Formula: see text] from the classical braid group [Formula: see text] into the automorphism group of the free group [Formula: see text], first described by Artin [Theory of Braids, Ann. Math. (2) 48(1) (1947) 101–126]. This homomorphism induces an action of [Formula: see text] on [Formula: see text] that can be recovered by considering the braid group as the mapping class group of [Formula: see text] (an upper half plane with [Formula: see text] punctures) acting naturally on the fundamental group of [Formula: see text]. Kauffman introduced virtual links [Virtual knot theory, European J. Combin. 20 (1999) 663–691] as an extension of the classical notion of a link in [Formula: see text]. There is a corresponding notion of a virtual braid, and the set of virtual braids on [Formula: see text] strands forms a group [Formula: see text]. In this paper, we will generalize the Artin action to virtual braids. We will define a set, [Formula: see text], of “virtual curve diagrams” and define an action of [Formula: see text] on [Formula: see text]. Then, we will show that, as in Artin’s case, the action is faithful. This provides a combinatorial solution to the word problem in [Formula: see text]. In the papers [V. G. Bardakov, Virtual and welded links and their invariants, Siberian Electron. Math. Rep. 21 (2005) 196–199; V. O. Manturov, On recognition of virtual braids, Zap. Nauch. Sem. POMI 299 (2003) 267–286], an extension [Formula: see text] of the Artin homomorphism was introduced, and the question of its injectivity was raised. We find that [Formula: see text] is not injective by exhibiting a non-trivial virtual braid in the kernel when [Formula: see text].
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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".