The tangled web of self-tying knots

Mathematician Leopold Kronecker stated “God created the integers, all else is the work of man,” alluding to the fact that the natural numbers most likely arose from physical counting, as in one's fingers or goats in the pasture. Topology, however, is arguably a different creature altogether and may have had its own independent origins from the physical world in the ubiquitous knot—something that cannot be undone without using the free ends because the individual strands cannot move through each other (1). One imagines a primordial knot getting tied accidentally in Cro-Magnon times and then tugged at to no avail … perhaps eventually cut. Of course, knots went on to have their uses in early societies, still far from any theoretical considerations but very much related to their ability to bind or secure things such as animals, sails, hair, etc. Knotted strings were also used by the Inca civilization for record-keeping and possibly even communication (2), still a few centuries before Euler began counting bridge-crossings in Königsberg.

The discovery and synthesis of polymers, long-chain molecules such as DNA, has brought a renewed physical relevance and context to knots (3, 4), and with it a new direction of study. For instance, it has been shown by the electrophoresis of loop DNA that knot types from the simplest trefoil to a knot with 10 crossings can occur at the molecular level (5). Although knots were actually tied recently in surfactant nanotubes by micromanipulation (6), molecular knots mostly occur in a spontaneous way, driven by competition between a fluctuating exploration of space due to Brownian motion and the excluded-volume effect (the string cannot pass through itself). Knots are a natural and sometimes irreversible result of this process, and despite scientific study, it is still true that “… a complete statistical mechanical description of …

*E-mail: belmonte{at}math.psu.edu