Learning knot invariants across dimensions
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Bibliographic record
Abstract
We use deep neural networks to machine learn correlations between knot invariants in various dimensions. The three-dimensional invariant of interest is the Jones polynomial J(q) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> , and the four-dimensional invariants are the Khovanov polynomial \text{Kh}(q,t) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="normal">Kh</mml:mtext> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> , smooth slice genus g <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>g</mml:mi> </mml:math> , and Rasmussen’s s <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>s</mml:mi> </mml:math> -invariant. We find that a two-layer feed-forward neural network can predict s <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>s</mml:mi> </mml:math> from \text{Kh}(q,-q^{-4}) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="normal">Kh</mml:mtext> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> with greater than 99&#37; <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>99</mml:mn> <mml:mi>%</mml:mi> </mml:mrow> </mml:math> accuracy. A theoretical explanation for this performance exists in knot theory via the now disproven knight move conjecture, which is obeyed by all knots in our dataset. More surprisingly, we find similar performance for the prediction of s <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>s</mml:mi> </mml:math> from \text{Kh}(q,-q^{-2}) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="normal">Kh</mml:mtext> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> , which suggests a novel relationship between the Khovanov and Lee homology theories of a knot. The network predicts g <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>g</mml:mi> </mml:math> from \text{Kh}(q,t) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="normal">Kh</mml:mtext> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> with similarly high accuracy, and we discuss the extent to which the machine is learning s <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>s</mml:mi> </mml:math> as opposed to g <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>g</mml:mi> </mml:math> , since there is a general inequality |s| &#8804;2g <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">|</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false" form="prefix">|</mml:mo> <mml:mo>≤</mml:mo> <mml:mn>2</mml:mn> <mml:mi>g</mml:mi> </mml:mrow> </mml:math>
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.004 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.014 |
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