Counting knots with fixed number of crossings

There are some known exponential bounds on the number. For example, if k_n is the number of prime knots with n crossings, then Welsh proved in "On the number of knots and links" (MR1218230) that

2.68 ≤ lim inf (k_n)^1/n ≤ lim sup (k_n)^1/n ≤ 13.5.

The upper bound holds if you replace k_n by the much larger number l_n of prime n-crossing links.

Sundberg and Thistlethwaite ("The rate of growth of the number of prime alternating links and tangles," MR1609591) also found asymptotic bounds on the number a_n of prime alternating n-crossing links: lim (a_n)^1/n exists and is equal to (101+√21001)/40.