Illuminating new and known relations between knot invariants

TL;DR

The paper addresses the problem of understanding how knot invariants relate across polynomial, homological, and hyperbolic classes by systematically testing correlations with neural networks on a large KnotInfo-derived dataset. It demonstrates that many known relationships can be recovered and uncovers novel correlations, notably between the Jones-type polynomials and hyperbolic longitude length $ll$, as well as links between Jones polynomial data and Floer homology invariants. It also proposes a conjectural, physics-inspired relation involving the colored Jones polynomials $J_n(e^{2\pi i /n};K')$, the hyperbolic volume $\,\mathrm{Vol}(K)$, and the longitude length via the formula $ \lim_{n \to \infty} \left( \log |J_n(e^{2\pi i /n};K')| - \frac{n}{2\pi} \mathrm{Vol}(K) \right) \approx \u001ell$, suggesting a generalized volume conjecture-like bridge between quantum algebraic data and hyperbolic geometry. The work demonstrates a data-driven pathway to explore deep mathematical structures, informs potential refinements of the FK correspondence, and motivates future unsupervised analyses to uncover intrinsic invariants-driven patterns. Overall, it provides a quantitative, experiment-driven map of how diverse knot invariants relate and hints at new theoretical directions in knot theory and mathematical physics. The results have potential to guide theoretical investigations into the connections between quantum invariants and geometric/topological attributes of knots.

Abstract

We automate the process of machine learning correlations between knot invariants. For nearly 200,000 distinct sets of input knot invariants together with an output invariant, we attempt to learn the output invariant by training a neural network on the input invariants. Correlation between invariants is measured by the accuracy of the neural network prediction, and bipartite or tripartite correlations are sequentially filtered from the input invariant sets so that experiments with larger input sets are checking for true multipartite correlation. We rediscover several known relationships between polynomial, homological, and hyperbolic knot invariants, while also finding novel correlations which are not explained by known results in knot theory. These unexplained correlations strengthen previous observations concerning links between Khovanov and knot Floer homology. Our results also point to a new connection between quantum algebraic and hyperbolic invariants, similar to the generalized volume conjecture.

Figures (9)

• Figure 1: Layerwise Relevance Propagation results from a neural network trained to learn $\varepsilon$ using evaluations of the Jones polynomial. Each column represents a knot, and every pair of rows represent the real and imaginary parts of the Jones polynomial evaluations at a particular point. Red pixels have higher relevance than blue pixels. Notice that certain evaluations are consistently more relevant than others. In this example, the final two rows are the most relevant. Although there are other rows that have high relevance for some knots, at least one of the final two rows (which correspond to a single evaluation) are highly relevant for every knot. These correspond to evaluations of the Jones polynomial at $t = -0.98+0.88i$.
• Figure 2: Heatmap showing accuracies of neural network predictions for $\varepsilon$ from single evaluations of the Jones polynomial. The horizontal axis is the real part of the point where the polynomial is evaluated, and the vertical axis is the imaginary part. The colouring shows the accuracy of the neural network predictions. Each point is averaged over five training runs. Only the upper half plane is included, since the Jones polynomial is holomorphic. The best-performing point is at $t = -0.6+0.1i$.
• Figure 3: Heatmap showing accuracies of neural network predictions for $\tau$ from single evaluations of the Jones polynomial. The horizontal axis is the real part of the point where the polynomial is evaluated, and the vertical axis is the imaginary part. Each point is averaged over five training runs. Only the upper half plane is included, since the Jones polynomial is holomorphic. The best-performing point is at $t = -0.7+0.1i$.
• Figure 4: Heatmap showing accuracies of neural network predictions for longitude length from single evaluations of the Jones polynomial. The horizontal axis is the real part of the point where the polynomial is evaluated, and the vertical axis is the imaginary part. Each point is averaged over five training runs. Only the upper half plane is included, since the Jones polynomial is holomorphic. One of the best-performing points is at $t= -1 + 0.2i$.
• Figure 5: Predictions from the neural network trained on an evaluation of the Jones polynomial at $t = -1 + 0.2i$. (Left) The real part of the Jones polynomial plotted against the longitude length. (Right) The imaginary part of the Jones polynomial plotted against the longitude length. The red points show the actual longitude length (darker red indicating higher density of points) and the green points show the neural network predictions.

Theorems & Definitions (1)

• Theorem 1