Big Data Approaches to Knot Theory: Understanding the Structure of the Jones Polynomial

Levitt, Jesse; Hajij, Mustafa; Sazdanovic, Radmila

doi:10.48550/arxiv.1912.10086

Cited by 2 publications

(2 citation statements)

References 23 publications

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“…Nevertheless, due to supersymmetric localization, there is a classical solution of the supersymmetric equations with the appropriate quantum numbers; what we cannot say is whether this solution is unique or if there are others which complicate the limit. 21 As well, [35,36] study many of the same invariants we do from dimensionality reduction and topological data analysis perspectives.…”

Section: Methodsmentioning

confidence: 99%

Learning knot invariants across dimensions

Craven¹,

Hughes²,

Jejjala³

et al. 2023

SciPost Phys.

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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)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.

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Section: Methodsmentioning

confidence: 99%

Learning knot invariants across dimensions

Craven¹,

Hughes²,

Jejjala³

et al. 2023

SciPost Phys.

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“…While trying to compare different approaches to building ensembles of closures for open arcs, we were struck by the similarity to a problem in machine learning: design a classifier which predicts the knot type of a closed polygon P given a subarc A of that polygon (for some existing machine learning approaches to knot classification, see [20,22,25,27,41]). In this paper, we will focus on equilateral closed polygons chosen from the uniform 1 probability measure [6,8] on such polygons.…”

Section: Introductionmentioning

confidence: 99%

Performance of the Uniform Closure Method for open knotting as a Bayes-type classifier

Tibor¹,

Annoni²,

Brine-Doyle³

et al. 2020

Preprint

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The discovery of knotting in proteins and other macromolecular chains has motivated researchers to more carefully consider how to identify and classify knots in open arcs. Most definitions classify knotting in open arcs by constructing an ensemble of closures and measuring the probability of different knot types among these closures. In this paper, we think of assigning knot types to open curves as a classification problem and compare the performance of the Bayes MAP classifier to the standard Uniform Closure Method. Surprisingly, we find that both methods are essentially equivalent as classifiers, having comparable accuracy and positive predictive value across a wide range of input arc lengths and knot types.

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Big Data Approaches to Knot Theory: Understanding the Structure of the Jones Polynomial

Cited by 2 publications

References 23 publications

Learning knot invariants across dimensions

Learning knot invariants across dimensions

Performance of the Uniform Closure Method for open knotting as a Bayes-type classifier

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