Signed Unknotting Number and Knot Chirality Discrimination via Strand Passage
Why this work is in the frame
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Bibliographic record
Abstract
Which chiral knots can be unknotted in a single step by a + to − (+−) crossing change, and which by a − to + (−+) crossing change? Numerical results suggest that if a knot with 6 or fewer crossings can be unknotted by a +− crossing change then it cannot be unknotted by a −+ one, and vice versa. However, we exhibit one chiral 8-crossing knot and one chiral 9-crossing knot which can be unknotted by either crossing change. Furthermore, we address the question analytically using results of Taniyama and Traczyk. We apply Taniyama's classification of unknotting operations to chiral rational knots and fully classify all those which, in a single step, can be unknotted by either type of crossing change; the first of these is 813. As a corollary, we obtain Stoimenow's result that all chiral twist knots can be unknotted by only one of the two crossing change types, +− or −+. Thus, as was observed numerically, all chiral knots with unknotting number one, and seven or fewer crossings, can be unknotted by only one of the two crossing change types. Traczyk's results allow us to address the question for some non-rational chiral unknotting number one knots with 9 or fewer crossings, however, for others the question remains open. We propose a numerical approach for investigating the latter type of knot. We also discuss the implications of our work in the context of DNA topology.
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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 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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 it