The title pretty much covers it: are there good (asymptotic) estimates on the number of knot types whose projection has at most $N$ crossings? Similar question with "projection" replaced by "alternating projection".



Number of (distinct) knots with a bounded number of crossings
Let $k(n)$ denote the number of prime knots with $n$ crossings, $l(n)$ the number of prime links with $n$ crossings, $a(n)$ the number of alternating prime links with $n$ crossings, and $ak(n)$ the number of prime alternating knots (all unoriented and unordered). 


Clearly from inclusion of sets $ak(n)\leq a(n)\leq l(n)$ and $ak(n)\leq k(n)\leq l(n)$. 


Then we have 


$$2.68 \leq \liminf\_{n\to \infty} k(n)^{\frac1n} \leq \liminf\_{n\to \infty} l(n)^{\frac1n} \leq 10.398...,$$ 


where the left is due to [Welsh](https://mathscinet.ams.org/mathscinet-getitem?mr=1218230) based on [Ernst-Sumners](https://doi.org/10.1017/S0305004100067323) (and only counts the growth of 2-bridge knots), and the right estimate is due to [Stoimenow](https://academic.oup.com/qjmath/article/55/1/87/1565728). 


For alternating knots, one has the same lower bound since 2-bridge knots are alternating. So one has 


$$2.68 \leq \liminf\_{n\to \infty} ak(n)^{\frac1n} \leq \lim\_{n\to \infty} a(n)^{\frac1n} = 6.14793...,$$ 
the right upper bound coming from [Sundberg-Thistlethwaite](https://mathscinet.ams.org/mathscinet-getitem?mr=1609591). 


From estimates on the growth of prime knots with $n$ crossings, one should be able to obtain upper bounds on the growth of all knots with $n$ crossings via the prime decomposition. Improved lower bounds over the prime growth are trickier, since we don’t know that crossing number is additive under connect sum.


