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We use deep neural networks to machine learn correlations betweenknot invariants in various dimensions. The three-dimensional invariantof 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 g g ,and Rasmussen’s s s -invariant.We find that a two-layer feed-forward neural network can predict s s from \text{Kh}(q,-q^{-4}) Kh ( q , − q − 4 ) with greater than 99% 99 % accuracy. A theoretical explanation for this performance exists in knottheory via the now disproven knight move conjecture, which is obeyed byall knots in our dataset. More surprisingly, we find similar performancefor the prediction of s s from \text{Kh}(q,-q^{-2}) Kh ( q , − q − 2 ) ,which suggests a novel relationship between the Khovanov and Leehomology theories of a knot. The network predicts g g from \text{Kh}(q,t) Kh ( q , t ) with similarly high accuracy, and we discuss the extent to which themachine is learning s s as opposed to g g ,since there is a general inequality |s| ≤2g | s | ≤ 2 g .The Jones polynomial, as a three-dimensional invariant, is not obviouslyrelated to s s or g g ,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) S U ( 2 ) Chern—Simons theory, and we review the gauge theory construction ofKhovanov homology which may be relevant for explaining the network’sperformance. 

In the Lambda-CDM model, dark energy is viewed as a constant vacuum energy density, the cosmological constant in the Einstein–Hilbert action. This assumption can be relaxed in various models that introduce a dynamical dark energy. In this paper, we argue that the mixing between infrared (IR) and ultraviolet (UV) degrees of freedom in quantum gravity leads to infinite statistics, the unique statistics consistent with Lorentz invariance in the presence of nonlocality, and yields a fine structure for dark energy. Introducing IR and UV cutoffs into the quantum gravity action, we deduce the form of Lambda as a function of redshift and translate this to the behavior of the Hubble parameter.