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Nature recently published a paper titled “Advancing mathematics by guiding human intuition with AI”. Using the power of linear algebra and calculus machine learning, the authors link "geometric" invariants of knots like the Chern-Simmons invariant and "algebraic" invariants like the Jones polynomial by proposing and proving a novel theorem:

$$|2 \sigma(𝐾) − \operatorname{slope}(K)|≤ c _1\operatorname{vol}(K)\operatorname{inj}(K)^{−3} + c_2$$

where $\operatorname{vol}(K)$ is the hyperbolic volume of a knot, and $\operatorname{slope}(K)=Re(\lambda/\mu)$ where $\lambda$ and $\mu$ are the longitudinal and meridional translations of $K$, respectively, and $c_1$, $c_2$ are constants independent of $K$. (The theorem is proven in The Signature and Cusp Geometry of Hyperbolic Knots.)

Here are some concrete questions (related to a subjective question I have which is not appropriate here):

Question 1: Does this theorem have any implications to established problems in knot theory? (The authors mention non-hyperbolic Dehn surgery.)

Question 2: Does the slope of a knot as defined above appear in other guises elsewhere in knot theory? (The authors claim no.)

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    $\begingroup$ The title seems completely different from the questions you ask in the body of the text. $\endgroup$
    – Sam Hopkins
    Commented Dec 6, 2021 at 15:38
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    $\begingroup$ For the question, it might be better to link the paper arxiv.org/abs/2111.15323 which contains more mathematical discussion of this result, the slope, and the signature. The title of this question could be changed to "The signature and cusp geometry of hyperbolic knots", the title of that paper, or "The signature and the slope of hyperbolic knots". I feel this title is going to be unnecessarily confusing if someone asks a (new) question about generalities of applying AI to mathematics. $\endgroup$
    – Will Sawin
    Commented Dec 6, 2021 at 16:02
  • $\begingroup$ I agree the original title was confusing. I changed the title to correspond to the paper being discussed. $\endgroup$
    – Ryan Budney
    Commented Dec 6, 2021 at 16:12
  • $\begingroup$ Looking at the result, the main thing that strikes me is the constraint on the slope entirely in the language of topological invariants. So you could think of this theorem as telling you a range of possible slopes on a cusped hyperbolic manifold, in order for this manifold to embed in $S^3$. I don't know if this theorem is better than you can do using other techniques, but that is the main feature that pops out at me. $\endgroup$
    – Ryan Budney
    Commented Dec 6, 2021 at 16:28
  • $\begingroup$ The original title was the name of the paper, though the new one makes the mathematical question clearer. $\endgroup$
    – Steve
    Commented Dec 6, 2021 at 19:21

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