Learning knot invariants across dimensions

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)J(q), and the four-dimensional invariants are the Khovanov polynomial \text{Kh}(q,t)Kh(q,t), smooth slice genus gg, and Rasmussen’s ss-invariant. We find that a two-layer feed-forward neural network can predict ss from \text{Kh}(q,-q^{-4})Kh(q,−q−4) with greater than 99%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 ss from \text{Kh}(q,-q^{-2})Kh(q,−q−2), which suggests a novel relationship between the Khovanov and Lee homology theories of a knot. The network predicts gg from \text{Kh}(q,t)Kh(q,t) with similarly high accuracy, and we discuss the extent to which the machine is learning ss as opposed to gg, since there is a general inequality |s| ≤2g|s|≤2g. The Jones polynomial, as a three-dimensional invariant, is not obviously related to ss or gg, but the network achieves greater than 95%95% accuracy in predicting either from J(q)J(q). Moreover, similar accuracy can be achieved by evaluating J(q)J(q) at roots of unity. This suggests a relationship with SU(2)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.