Bounds on the number of knots with a fixed number of crossings

Question

Let $K(n)$ denote the number of distinct knots on $n$ crossings and let $A(n)$ be the number of alternating knots on $n$ crossings. What are some upper and lower bounds for $K(n)$ and $A(n)$?

Answer 1

Well, this is probably too late, but here we go. I found this question while looking for an answer myself. If someone happens upon a better bound for $K(n)$, I would hope that gets posted here eventually too.

Using the The rate of growth of the number of prime alternating links and tangles by Sundberg and Thistlethwaite (1998), we get the following, which I am just going to screenshot and add since it is a lot to type. But we have bounds on $A(n)$, which of course, bounds $K(n)$ below.

In my mind, the quick take away, is that alternating knots are growing at about $6.147^n$.