Bibliographic record
Abstract
A new topological operad is introduced, called the splicing operad. This operad acts on a broad class of spaces of self-embeddings N→N, where N is a manifold. The action of this operad on EC(j, M) (self-embeddings ℝj×M→ ℝj×M with support in Ij×M) is an extension of the action of the operad of (j+1)-cubes on this space defined in Budney [‘Little cubes and long knots’, Topology 46 (2007) 1–27]. Moreover, the action of the splicing operad encodes a version of Larry Siebenmann's [Bonahon and Siebenmann, ‘New Geometric Splittings of Classical Knots, and the Classification and Symmetries of Arborescent Knots’, Preprint; Siebenmann, ‘On vanishing of the Rohlin invariant and nonfinitely amphicheiral homology 3-spheres’, Proc. Sympos., Univ. Siegen, Siegen, 1979, Lecture Notes in Math. 788 (Springer, Berlin, 1980) 172–222] splicing construction for knots in S3 in the j=1, M=D2 case, for which we denote the splicing operad SP3, 1. The space of long knots in ℝ3 (denoted by K3, 1) was shown to be a free algebra over the 2-cubes operad with free generating subspace P ⊂ K3, 1, the subspace of long knots that are prime with respect to the connect-sum operation [R. Budney, ‘Little cubes and long knots’, Topology 46 (2007) 1–27]. One of the main results of this paper is that K3, 1 is free with respect to the splicing operad SP3, 1 action, but the free generating space is the significantly smaller space of torus and hyperbolic knots T H⊂K3, 1. Moreover, we show that SP3, 1 is a free product of two operads. The first free summand of SP3, 1 is a semi-direct product w operad which is not equivalent to the framed discs operad. The second free summand of SP3, 1 is a free Σ ι O 2 -operad, free on Σ ι O 2 -spaces that encode cabling and hyperbolic satellite operations; moreover, the Σ ι O 2 -homotopy type of these spaces is determined by finding adapted maximal symmetry positions for hyperbolic links in S3. This is an in-principle explicit description of the homotopy type of the space of knots in S3, and modulo the rather difficult problem of determining the symmetry groups of a class of hyperbolic links and their actions on the cusps, this is a closed form of description of the homotopy type.
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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 | 0.000 |
| 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".