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)
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"). There are lower exponential bounds 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.