Sunday, March 1, 2009

Knot Interesting

We all know that knots form in christmas tree lights or ipod earphone cords spontaneously. 

I thought that the longer the string, the greater the chance of knotting. It turns out that this is not true. The chance of knotting plateaus after a while. At right we have real data from Raymer and Smith at UCalf-San Diego. They shook pieces of rope in a box and counted the knots. They are mathematicians so they topologically categorized the knots. 

The intuitive notion is that there are only two rope ends, and as the rope gets longer, the number of ends stay the same. Any knot must start at the end.

In polymer physics there is the notion of reptation theory which describes the motion of a long linear polymer in the midst of other long linear polymers, as a snake slithering though a snake-pit like in Indiana Jones. Key is the prediction that the motion scales with the molecular weight cubed, which does not fit well with out knot theory.

The Raymer/Smith experiment contradicts the Frisch-Wasserman-Delbruck conjecture which says the chance of a polymer chain being knotted goes to one as the chain increases. And in fact, this has been proven mathematically using a random walk model. This has practical importance in electrophoretic separations. 

I think that the difficulty with reconciling the experiment and the theory is that the infinitely long chain needs an vastly longer time to become knotted. In a practical experiment with fixed tumbling time, there would be a plateau effect. However, at molecular scales polymer probably get tangled faster than common ropes. Secondly, one could speed tangling and knotting by raising the temperature via a time-temperature superposition effect. Third, the theoreticians were considering a rope on a large lattice, and the practical experiment was in a box of fixed size, where longer ropes took up a greater fraction of the free volume -- restricting rope movement. 

An interesting variation is a T-shaped rope. The extra end should increase the number of knots by at least 50%. On the other hand, the decrease in viscosity for T-shape (and higher star chains) is well established. A second contradiction. 

Of course my ipod earphone cords are actually T-shaped, and the theory of T-shaped chains has some practical appeal. I have taken to hanging my earphones instead of coiling them.