  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.
