Balanced Presentations of the Trivial Group and 4-dimensional Geometry
Bibliographic record
Abstract
We construct a sequence of balanced presentations of the trivial group with two generators and two relators with the following property: The minimal number of relations required to demonstrate that a generator represents the trivial element grows faster than the tower of exponentials of any fixed height of the length of the finite presentation. We prove that 1) There exist infinitely many non-trivial codimension one "thick" knots in R5; 2) For each closed four-dimensional smooth manifold M and for each sufficiently small positive ε the set of isometry classes of Riemannian metrics with volume equal to 1 and injectivity radius greater than ε is disconnected; and 3) For each closed four-dimensional PL-manifold M and any m there exist arbitrarily large values of N such that some two triangulations of M with < N simplices cannot be connected by any sequence of < expm( N) bistellar transformations, where expm( N) = exp(exp(...exp(N))) (m times). We construct families of trivial 2-knots Ki in R4 such that the maximal complexity of 2-knots in any isotopy connecting Ki with the standard unknot grows faster than a tower of exponentials of any fixed height of the complexity of Ki. Here we can either construct Ki as smooth embeddings and measure their complexity as the ropelength (a.k.a the crumpledness) or construct PL-knots Ki, consider isotopies through PL knots, and measure the complexity of a PL-knot as the minimal number of at 2-simplices in its triangulation. For any m we produce an exponential number of balanced presentations of the trivial group with four generators and four relations of length N such that the minimal number of Andrews-Curtis transformations needed to connect any two of the presentations is at least expmi(N).
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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 | 0.000 | 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) | 0.004 | 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".