Learning knot invariants across dimensions
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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 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 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 Kh(q,-q^-2), which suggests a novel relationship between the Khovanov and Lee homology theories of a knot. The network predicts g from 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| ≤ 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.
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