3 edited tags
2 clarified crossing #; deleted 1 characters in body

Connect $n$ random points on a sphere in a cycle of segments between succesive points:

I would like to know the growth rate, with respect to $n$, of the crossing number (the minimal number of crossings of any diagram of the knot) $c(n)$ of such a knot. I only know that $c(n)$ is $O(n^2)$, because it is known that the crossing number is upper-bounded by the stick number $s(n)$: $$\frac{1}{2}(7+\sqrt{ 8 c(K) + 1}) \le s(K)$$ for any knot $K$. And $s(n) \le n$ is immediate.

I feel certain this has been explored but I am not finding it in the literature. Thanks for pointers!

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# Complexity of random knot with vertices on sphere

Connect $n$ random points on a sphere in a cycle of segments between succesive points:

I would like to know the growth rate, with respect to $n$, of the crossing number $c(n)$ of such a knot. I only know that $c(n)$ is $O(n^2)$, because it is known that the crossing number is upper-bounded by the stick number $s(n)$: $$\frac{1}{2}(7+\sqrt{ 8 c(K) + 1}) \le s(K)$$ for any knot $K$. And $s(n) \le n$ is immediate.

I feel certain this has been explored but I am not finding it in the literature. Thanks for pointers!