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Has there been any research into something like the ratio of distinct Alexander-indistinguishable knots to total knots (up to some measure of complexity)? This was a random question asked of me by a student, and I have no idea (though my gut feeling is that this is far too much to ask).

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    $\begingroup$ I suppose it depends on your sense of complexity. If you use something geometric that is readily relateable to the Alexander polynomial (say, JSJ decompositions) then there are simple answers: usually there are infinitely many Alexander-indistinguishable knots for "most" knots. But if your sense of complexity is one that's difficult to relate to geometry (like crossing number) then there's little known. $\endgroup$ – Ryan Budney Oct 11 '18 at 4:16

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