Pardon the naive question. Perhaps there are several different variants of my question, but essentially I am seeking to know whether ribbon knots are vanishingly rare among all knots, or a positive fraction of all knots. I am sure this is known to the cogknotscenti...       PrimeKnots

If the question needs to be specific:

Q. Among prime knots of crossing number $n$ (OEIS A002863), what is the fraction that are ribbon knots, as $n \to \infty$?

  • 8
    $\begingroup$ Surely they are vanishingly rare. The Fox-Milnor condition on the Alexander polynomial (that it factors as $f(t)f(t^{-1})$) should already show that slice knots are rare. I don't know a proof, however. $\endgroup$ – Dylan Thurston Nov 16 '14 at 3:44
  • $\begingroup$ @DylanThurston: Thanks for pointing me to that nice symmetry condition on ribbon (and slice) knots. $\endgroup$ – Joseph O'Rourke Nov 16 '14 at 13:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.