What is the asymptotics (in L) for the number of topologically different knots possible using a perfectly flexible, non-selfintersecting rope of length L and radius 1? (With ends glued together after knotting)



Growth of knots possible with rope of length L
I believe this is an open question. There are some possible estimates.


Let $cr(K)$ denote the crossing number of $K$. There are upper and lower bounds on the ropelength in terms of crossing number (see the section "[Dependence of ropelength on other knot invariants](http://en.wikipedia.org/wiki/Ropelength)"). There are [lower exponential bounds](http://www.ams.org/mathscinet-getitem?mr=1218230) on the number of prime knots $K$ with $cr(K)\leq n$, so in principle this translates into upper and lower exponential bounds for the number of knots with a given thickness (the upper bound may be made crudely in terms of the number of 4-valent $n$-vertex planar graphs multiplied by $2^n$). However, the exponents in the upper and lower bounds are different, so this is far from giving asymptotics. 


