Learning knot invariants across dimensions

TL;DR

This paper demonstrates that neural networks can accurately predict complex knot invariants across dimensions, revealing potential new relationships and connections to gauge theory, with implications for understanding knot theory through machine learning.

Contribution

The study shows neural networks can predict knot invariants like the Rasmussen s-invariant and slice genus with high accuracy, uncovering novel links between different knot invariants and theories.

Findings

Neural networks predict s-invariant from Khovanov polynomial with >99% accuracy.

Networks predict slice genus g from Khovanov polynomial with high accuracy.

Prediction of invariants from Jones polynomial suggests links to Chern--Simons theory.

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 $J(q)$, 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.