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What is the asymptotic growth in L for the numer of topological different knots possible using a length L closed string of radius 1?

In 3 dimension euclidean space.

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Let $Cr(K)$ be the crossing number of a knot $K$. If $L(K)$ is the minimal length of a knot $K$ with thickness 1, it has been shown that $L(K)=\Omega(Cr(K)^{3/4})$, first proved in this paper by Gregory Buck Hopefully this is close enough to the answer that you needed. See also

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