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As far as I know, there is a classification of all prime knots with less than 16 crossings.

It seems that there is already a fast enough algorithm to distinguish a knot from an unknot.

So in principle there is a huge amount of data to implement a deep learning machine which will recognize (and distinguish) knots up to some very good accuracy.

Is it something that mathematicians have tried to do? Any references?

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    $\begingroup$ I am unaware of any. What would the purpose of such an algorithm be? The algorithms for knot recognition are fast and "small" enough that you could implement them on most tablets. I suppose it would be interesting to see what kinds of correct and incorrect inferences a deep learning algorithm would make, but that would mostly inform us on the strengths and flaws of that particular deep learning setup. Perhaps a subtle issue your deep learning algorithm needs to grapple with, is what is the input to the algorithm, i.e. how would it think about knots? $\endgroup$
    – Ryan Budney
    Commented Nov 17, 2021 at 16:50
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    $\begingroup$ Another issue deep learning would have to grapple with is that while there are an enormous number of classified knots, it isn't a very "fair" sampling of knots, from various perspectives. The tables are made according to crossing number, not, for example, by geometrization. So there aren't very many sattelite knots in the table, even though technically those are "most" knots, from many useful points of view. $\endgroup$
    – Ryan Budney
    Commented Nov 17, 2021 at 16:53
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    $\begingroup$ There's even a classification of prime knots with up to 19 crossings: regina-normal.github.io/data.html $\endgroup$
    – Lukas Lewark
    Commented Dec 22, 2021 at 7:56

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I saw two articles today (12/2/21) that reminded me of this post. I am mentioning them here to potentially help the OP:

  1. Learning knot invariants across dimensions by Jessica Craven, Mark Hughes, Vishnu Jejjala, Arjun Kar (on arXiv)
  2. DeepMind’s AI helps untangle the mathematics of knots, by Davide Castelvecchi (on nature.com).
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Adding to Sean Lawton's answer, there was an arxiv posting yesterday by Davies-Juhász-Lackenby-Tomasev, The signature and cusp geometry of hyperbolic knots that describes a relation between the cusp geometry of a hyperbolic knot, its signature, and its hyperbolic volume. The authors describe how this was found via machine learning.

This is not exactly what the OP asked for (it doesn't speak specifically to classification) but in many ways seems like a more interesting use of technology to discover heretofore unseen relationships.

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You may want to take a look at this article . The article is both entertaining and well written, and here are its core ideas:

  1. KNOTS AS WORDS IN A LANGUAGE

First core idea is to leverage the representation of knots as phrases in a suitable language. This is the cool part of the article, and actually a good recap of the fundamentals of Knot Theory, so I shall say no further. Have fun!

  1. KNOT WORD EMBEDDING

There are many tools available for handling languages in deep learning. The key point is that one can learn how to embed words, phrases, documents into a vector space, in such a way as to preserve its contextual meaning. The first and most famous is word2vec, which is a shallow embedding. But now there is an entire artillery of nural tools which do the embedding at the appropriate level of sofistication, for instance BERT. This step is needed because, after all, deep learner can "eat" only tensors of numbers. Moreover, the embedding, if done right, reduces the dimensionality of the input, and third because the good embedding preserves some contextual information, which can then be leveraged by the downstream classifier

  1. CLASSIFYING KNOTS

At this point every knot is a low dim vector, and you want to classify it. As far as I understand, only basic classification is attempted, namely between knots that can be unknotted and the others (though I think this machinery can be expanded much further).

In the article a Reinforcement Learning paradigm which requires creating a suitable set of examples of the two categories is chosen for the binary classification task. My thought is that one should also explore Adversarial Neural Networks o the same purpose (basically one is the Unknotter and the other is the Cheater, he sends knots that look like they are unknottable but they are not)

Don't know of any concrete implementation, but I would be surprised if something out of this paper is not to be found in github.

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