Spontaneous knotting of an agitated string

Abstract

It is well known that a jostled string tends to become knotted; yet the factors governing the ''spontaneous'' formation of various knots are unclear. We performed experiments in which a string was tumbled inside a box and found that complex knots often form within seconds. We used mathematical knot theory to analyze the knots. Above a critical string length, the probability P of knotting at first increased sharply with length but then saturated below 100%. This behavior differs from that of mathematical self-avoiding random walks, where P has been proven to approach 100%. Finite agitation time and jamming of the string due to its stiffness result in lower probability, but P approaches 100% with long, flexible strings. We analyzed the knots by calculating their Jones polynomials via computer analysis of digital photos of the string. Remarkably, almost all were identified as prime knots: 120 different types, having minimum crossing numbers up to 11, were observed in 3,415 trials. All prime knots with up to seven crossings were observed. The relative probability of forming a knot decreased exponentially with minimum crossing number and Mö bius energy, mathematical measures of knot complexity. Based on the observation that long, stiff strings tend to form a coiled structure when confined, we propose a simple model to describe the knot formation based on random ''braid moves'' of the string end. Our model can qualitatively account for the observed distribution of knots and dependence on agitation time and string length.Jones polynomial ͉ knot energy ͉ knot theory ͉ random walk ͉ statistical physics K nots have been a subject of scientific study since as early as 1867, when Lord Kelvin proposed that atoms might be described as knots of swirling vortices (1). Although this theory fell into disfavor, it stimulated interest in the subject, and knots currently play a role in many scientific fields, including polymer physics, statistical mechanics, quantum field theory, and DNA biochemistry (2, 3). Knotting and unknotting of DNA molecules occurs in living cells and viruses and has been extensively studied by molecular biologists (4-6). In physics, spontaneous knotting and unknotting of vibrated ball-chains have recently been studied (7-9). In mathematics, knot theory has been an active field of research for more than a century (3).Formation of knots in mathematical self-avoiding random walks has been extensively studied (10-16). In the 1960s, Frisch and Wasserman (10) and Delbruck (11) conjectured that the probability of finding a knot would approach 100% with an increasing walk length. In 1988, Sumners and Whittington (15) proved this conjecture rigorously by showing that exponentially few arcs would remain unknotted as the length tends to infinity. Numerical studies of finite-length random walks find that the probability of knotting and the average complexity of knots increase sharply with the number of steps (16).Here, we describe a simple physical experiment on knot formation. A string was placed in a cubic box and the...

Untying Knotted DNA with Elongational Flows

“…We next explore the convection-dominated regime, i.e., K ≫ l, in greater detail for the following topologies: 3 1 , 4 1 , 5 1 , 5 2 , 6 1 , 6 3 , 7 1 , 9 1 , 10 28 , 11 1 , and 15n165258. 42 The mean displacements of these knots are plotted in Figure 3a.…”

“…1 Formally defined only for closed rings, the topologies of "open" knots (referred to hereafter simply as knots) are often unambiguous (e.g., shoelaces and neckties) and can be closed and algorithmically defined. 2−4 At microscopic scales, chromosomal knots are modified by topoisomerases during cell division 5 and are thought to participate in gene regulation.…”

The tangled web of self-tying knots

“…Of course, it seems reasonable that the longer the string, the more likely it will tie itself up, but surprisingly, they find that the probability of knotting stops increasing at some length; this was also observed independently for a bouncing chain (20). The reasons for this resistance to knotting are dynamic: the string was too stiff or otherwise did not have enough room to move about in the box (8). This may have implications for the knottedness of confined DNA.…”

“…For instance, there is only one open knot realization of the 6 1 but several for the 6 3 . These open knots are closer to what can tie and untie (in a string or chain) (14,18) and what was studied by Raymer and Smith (8).…”

“…It is at these points that the tension decreases because of the friction between the two parts of the string (12,13), allowing for the main technological importance of certain knots: they hold. Exactly which knots do or do not slip off not only determines their utility but also plays a role in the probability of finding a particular knot tied spontaneously (8,14,15).…”