Bibliographic record
Abstract
We give examples of knots distinguished by the total rank of their Khovanov homology but sharing the same two-fold branched cover.Hence Khovanov homology does not yield an invariant of two-fold branched covers.Mutation provides an easy method for producing distinct knots sharing a twofold branched cover: The mutation in the branch set corresponds to a trivial surgery in the cover.Due to a result of Wehrli [2007; 2009] (see also [Bloom 2009]), this provides a range of examples of manifolds that branch cover S 3 in more than one way, but for which the distinct branch sets have identical rank in their respective Khovanov homology groups over ކ 2 = .ޚ2/ޚFrom this point of view this fact is not completely surprising, as Khovanov homology is closely related to the Heegaard Floer homology of two-fold branched covers [Ozsváth and Szabó 2005].Indeed, this is made precise in Bloom's proof of mutation invariance [2009].More generally, there is a question posed by Ozsváth: Is Khovanov homology an invariant of the two-fold branched cover?More precisely, is the total rank of the reduced Khovanov homology (over ކ 2 ) an invariant of two-fold branched covers?This short note gives a negative answer.Theorem.The total rank of Khovanov homology is not an invariant of two-fold branched covers.This theorem is proved by exhibiting manifolds that are two-fold branched covers of S 3 in two different ways, and for which the pair of branch sets is distinguished by the total rank in Khovanov homology.We work with the reduced version of Khovanov homology, denoted Kh, with ކ 2 coefficients [Khovanov 2000;2003].Surgery on torus knots.Let S 3 r/s (K ) denote the result of (r/s)-surgery on a knot K ↩→ S 3 , and let T p,q denote the positive ( p, q) torus knot in S 3 (with 0 < p < q).Note that, as we will only consider torus knots, p and q are relatively prime.Proposition 1 [Moser 1971].The manifold S 3 ±1/n (T p,q ) is Seifert fibered with base orbifold S 2 ( p, q, pqn ∓ 1) for n > 0.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.008 | 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".