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".

up vote 13 down vote accepted

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 based on Ernst-Sumners (and only counts the growth of 2-bridge knots), and the right estimate is due to Stoimenow.

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.

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.

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