A mnemonic for the Lipshitz–Ozsváth–Thurstoncorrespondence
Bibliographic record
Abstract
When k is a field, type D structures over the algebra kŒu; v=.uv/ are equivalent to immersed curves decorated with local systems in the twice-punctured disk.Consequently, knot Floer homology, as a type D structure over kŒu; v=.uv/, can be viewed as a set of immersed curves.With this observation as a starting point, given a knot K in S 3 , we realize the immersed curve invariant c HF.S 3 X V .K// of Hanselman, Rasmussen and Watson by converting the twice-punctured disk to a once-punctured torus via a handle attachment.This recovers a result of Lipshitz, Ozsváth and Thurston calculating the bordered invariant of S 3 X V .K/ in terms of the knot Floer homology of K. 57K18, 57K31; 57R58Recent work interprets relative versions of homological invariants in terms of immersed curves, including Heegaard Floer homology for manifolds with torus boundary (see Hanselman, Rasmussen and Watson [4]) as well as link Floer homology (see Zibrowius [23]), singular instanton knot homology (see Hedden, Herald and Kirk [7]), and Khovanov homology (see Kotelskiy, Watson and Zibrowius [12]) for 4-ended tangles.In particular, Section 5 of [12] classifies type D structures over a quiver algebra associated with a surface with boundary in terms of immersed curves on this surface; compare Haiden, Katzarkov and Kontsevich [2] and Hanselman, Rasmussen and Watson [4].Denoting a field by k, perhaps the simplest algebra to illustrate these classification results is R D kŒu; v=.uv/.This algebra arises as the path algebra of a quiver that is associated with the decorated surface shown in Figure 1.Work of Lekili and Polishchuk [13;14] describes the role of R, and its relationship with the twice-punctured disk, in the context of homological mirror symmetry; see in particular [14, Figures 1 and 2].The algebra R equipped with the Alexander and ı gradings gr.u/ D .1; 1/ and gr.v/ D .1;1/ plays a central role in knot Floer homology; see Dai, Hom, Stoffregen and Truong [1], for instance.
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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 | 0.002 |
| 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 | 0.002 | 0.006 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.016 | 0.003 |
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".