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Learning knot invariants across dimensions
by Jessica Craven, Mark Hughes, Vishnu Jejjala, Arjun Kar
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Jessica Craven · Vishnu Jejjala |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2112.00016v2 (pdf) |
| Date accepted: | 2022-11-11 |
| Date submitted: | 2022-10-26 07:30 |
| Submitted by: | Craven, Jessica |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
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 $J(q)$, and the four-dimensional invariants are the Khovanov polynomial $\text{Kh}(q,t)$, smooth slice genus $g$, and Rasmussen's $s$-invariant. We find that a two-layer feed-forward neural network can predict $s$ from $\text{Kh}(q,-q^{-4})$ with greater than $99\%$ 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 $s$ from $\text{Kh}(q,-q^{-2})$, which suggests a novel relationship between the Khovanov and Lee homology theories of a knot. The network predicts $g$ from $\text{Kh}(q,t)$ with similarly high accuracy, and we discuss the extent to which the machine is learning $s$ as opposed to $g$, since there is a general inequality $|s| \leq 2g$. The Jones polynomial, as a three-dimensional invariant, is not obviously related to $s$ or $g$, but the network achieves greater than $95\%$ accuracy in predicting either from $J(q)$. Moreover, similar accuracy can be achieved by evaluating $J(q)$ at roots of unity. This suggests a relationship with $SU(2)$ Chern--Simons theory, and we review the gauge theory construction of Khovanov homology which may be relevant for explaining the network's performance.
List of changes
On page 5, $V$ is intended to be a general $\mathbb{Q}$-vector space while $W$ is an example. We have modified the language to clarify this point.
The typo ``for for'' has been corrected.
Figure~3 on page~20 and Figure~4 on page 26 now have the same $y$-axis to facilitate the comparison.
In Section~4.5.1, we have added the following sentences: ``Since the slice genus and the $s$-invariant are closely related, we are not sure whether the networks are learning the $s$-invariant and computing $g$ from $s$, learning $g$ and computing $s$ from $g$, or learning both invariants independently. Using a multi-task network could help to determine the most plausible option. For example, if $\mathcal{W}_s > \mathcal{W}_g$ and the network learns both invariants well, this suggests that the neural networks might be learning $s$ and learning $g$ via $s$. However, if the performance drops, then we may conclude that learning $g$ is the more natural task.''
Reference~[54] is added with a link to a GitHub repository.
Published as SciPost Phys. 14, 021 (2023)