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 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 | 0.006 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.010 | 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".