In this really cool paper https://arxiv.org/abs/1612.03368, A. Malyutin shows that the probability that a random prime knot of up to $N$ crossings (as $N$ goes to infinity) is not generically hyperbolic (mod some "standard conjectures"). Now, the question is, how would one go about generating prime knots, either in order of crossing number (preferred) or in some randomized way? It is an interesting question of which conjectures are actually not correct (since, just like HRJW in the comments, I believe that a random prime knot should be hyperbolic).
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asked Mar 23, 2018 at 1:32
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$\begingroup$ Dunfield has an approach to producing random prime knots, with some experimental data: faculty.math.illinois.edu/~nmd/slides/random_knots.pdf $\endgroup$– Ian AgolCommented Mar 23, 2018 at 4:19
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$\begingroup$ @IanAgol Interesting, though he seems to basically use the rejection method, which has numerous issues. $\endgroup$– Igor RivinCommented Mar 23, 2018 at 4:25
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4$\begingroup$ Could you clarify what you assert that Malyutin's paper actually says? There seems to be a typo in your first sentence. But a quick glance suggests that he only proves that "a random prime knot ... is not generically hyperbolic" conditional on various other conjectures. Surely one might equally believe that a random prime knot is generically hyperbolic, but that the other conjectures are false? $\endgroup$– HJRWCommented Mar 23, 2018 at 8:54
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3$\begingroup$ @HRJW I am not sure what to believe (though I tend to agree with your point of view, as you probably know), which is why I am thinking of how to design an experiment, hence the question... $\endgroup$– Igor RivinCommented Mar 23, 2018 at 13:08
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I'm not sure this is what you are looking for, but Adams et al. (https://arxiv.org/abs/1208.5742) show how every knot is obtained from a permutation in $S_n$. We sampled random permutations and used SnapPy to determine the resulting knot types, and they seem to be generically hyperbolic (https://arxiv.org/abs/1711.10470, Figure 14).
