Jessica Craven, Mark Hughes, Vishnu Jejjala, Arjun Kar
SciPost Phys. 14, 021 (2023) ·
published 21 February 2023

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We use deep neural networks to machine learn correlations between knot invariants in various dimensions. The threedimensional invariant of interest is the Jones polynomial $J(q)$, and the fourdimensional invariants are the Khovanov polynomial $\text{Kh}(q,t)$, smooth slice genus $g$, and Rasmussen's $s$invariant. We find that a twolayer feedforward 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 threedimensional 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 $SU(2)$ Chernâ€”Simons theory, and we review the gauge theory construction of Khovanov homology which may be relevant for explaining the network's performance.
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