More knots as crossing number increases?

Question

Reading knot tables, it seems that as $n$ increases, more prime knots have crossing number $n$. Is this a proven fact? More precisely,

If $k(n)$ is the number of knots with crossing number $n$, is $k(n)$ strictly increasing?

Asymptotic results are good to know but I'm interested in absolute increase; it would be rather intriguing if $k(m) \leq k(m + 1)$ for some $m$.

If (strict) increasingness has been proven, can I have a reference to the proof?

What about if $k(n)$ is restricted to prime knots, and/or mirror images are ignored?

Answer 1

While it is true that the number of knots of a given crossing number increases really fast, the answer in general is not known. See for example http://en.wikipedia.org/wiki/Knot_theory#Tabulating_knots and the reference (I believe you can find the article on arxiv).

The problem basically is that very little is known about the crossing number of a knot: any reasonable property that it might have is still conjectural. For example: is it true that $\operatorname{cr}(K\#K')= \operatorname{cr}(K)+\operatorname{cr}(K')$? There are a few results in this context, and overhelming evidence for the validity of the formula, but still no proof in sight.

Answer 2

The sequence of prime knots does indeed grow. This sequence can be seen on OEIS.

Various asymptotic formulae are known for this and related sequences based on the crossing number, but proofs are difficult. For instance, it is conjectured that the number of prime alternating knots with the crossing number $n$ is

$$d_n \sim \eta n^{\xi} \kappa^n$$

with

$$\xi = \frac{\sqrt{13} + 1}{6} - 3$$

and $\kappa$ and $\eta$ some constants whose relation to other constants is still murky. Although this is still (I believe, as of several years back at least) conjectured, there are known asymptotes for links and tangles, so there are lower limits that answer your question in the affirmative.

See, for example, "The asymptotic number of prime alternating links" by Kunz-Jacques and Schaeffer or "The rate of growth of the number of prime alternating links and tangles" by Sundberg and Thistlewaite.