The entropic cost of tying knots in polymers

The idea of knot inflation is that of A. Grosberg and it is tricky, but I'll try to explain some of it. Construction by way of scaling analysis is required to tackle this.

Suppose the volume of our polymer is given by $R$. Now fit a tube of fixed length, $L$, over the polymer; this tube has diameter, $D$, across its length that is just large enough to contain the polymer. Now the tube is inflated with a fixed $L$, then $D$ is uniformly allowed to increase until a maximum $D_{max}$ is reached. An aspect ratio, $p=L/D_{max}$, is then a weak topological invariant. Affine transformation of the inflated tube is done so that the polymer will fit in a sphere of size $R$, with $L_R$ and $D_R$ such that $L_R/D_R=p$ and $L_R D_R^2 \approx R^3$. He then uses this construction for finding the entropy.

The trick now is to assume that most of the available conformations of our polymer with a certain knot are also available to a trivial knot within our sphere of size $R$, giving $$S= S_{trivial} + \Delta S_{approx}$$

We can say that $S_{trivial}$ should be proportional to $N$ and so we can rewrite it as, $$S = sN + \Delta S_{approx}$$

with $s$ not a function of $p$ or $N$.

Now he confines an ideal knot in a cylinder (ties the polymer to both ends of the cylinder), and considers what will happen to the entropy if $p$, of the cylinder, is varied. He finds that $\Delta S_{approx}$ is proportional to $-p$. Finally, a polymer with the maximum amount of knots should give $S\approx 0$, so that means $p \approx N$ in that case, giving $$S = s(N-p)$$

Thus the entropy decreases with increasing knot complexity, $p$. This method is very crude, at best, but it does provide a qualitative approach to your query.

To see his full argument I'll have to refer you to the citation: A.Y. Grosberg “Entropy of a Knot: Simple Arguments About Difficult Problem,” In: Ideal Knots, Edited by A.Stasiak, V.Katrich, L.H.Kauffman, World Scientific, 1998, p. 129-142.

Addendum:

In that same article he extends his argument to calculate the probability of finding a knot as $$P(N) = N_0 e^{-N/N_0 - sp} $$

I asked this related question on mathoverflow: https://mathoverflow.net/questions/72937/what-are-the-statistics-of-prime-knots-in-3d-random-walk. The physics references provided in the answer, and the articles which cite them, are illuminating.