Bibliographic record
Abstract
It had been known since old times (works of Murakami–Ohtsuki, Cheptea–Le and the second author) that there exists a universal finite type invariant ("an expansion") Z old for knotted trivalent graphs (KTGs), and that it can be chosen to intertwine between some of the standard operations on KTGs and their chord-diagrammatic counterparts (so that relative to those operations, it is "homomorphic"). Yet perhaps the most important operation on KTGs is the "edge unzip" operation, and while the behavior of Z old under edge unzip is well understood, it is not plainly homomorphic as some "correction factors" appear. In this paper we present two equivalent ways of modifying Z old into a new expansion Z, defined on "dotted knotted trivalent graphs" (dKTGs), which is homomorphic with respect to a large set of operations. The first is to replace "edge unzips" by "tree connected sums", and the second involves somewhat restricting the circumstances under which edge unzips are allowed. As we shall explain, the newly defined class dKTG of KTGs retains all the good qualities that KTGs have — it remains firmly connected with the Drinfel'd theory of associators and it is sufficiently rich to serve as a foundation for an "algebraic knot theory". As a further application, we present a simple proof of the good behavior of the LMO invariant under the Kirby II (band-slide) move, first proven by Le, Murakami, Murakami and Ohtsuki.
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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 | 0.002 | 0.003 |
| 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".