This is a second question about statistical geometry. The image https://commons.wikimedia.org/wiki/File:Ball_of_yarn_10.jpg shows a typical ball of yarn. The idealized yarn of the question is assumed to be unstretchable and infinitely flexible.
If I take an X-ray of an average ball of yarn (ball radius $R$, yarn radius $r$), how may crossings will I see, on average?
Probably (but I am not sure), the total length will go like $R^3/r^2$. What can be said about the crossing count?