INTRODUCTION
The practical value of knots has long been recognised, with indirect evidence of their use dating back to the time of Homo erectus 300,000 years ago1 . However, the apparent tendency of anything capable of forming a knot to do so is equally well-attested. The potentially lethal ability of a single knot to reduce the breaking strength of climbing rope by up to 50 per cent2 has been noted since the earliest days of professional mountaineering3, prompting great care to be taken in the transport and storage of ropes. Forensic scientists are also wary of interpreting the significance of knots found at the scene of crime, following recognition that knots do seem capable of forming "by themselves"4.
Given the determined effort usually needed to create knots, their apparently spontaneous generation is certainly puzzling. Indeed, there seems no obvious way to explain how rope, flex etc. can go through precisely the contortions needed to form a knot - apart from, of course, the essential cussedness of the universe captured by Murphy's Law: "If something can go wrong, it will". As with so many notorious manifestations of this "Law", from toast landing butter side down to the plethora of odd socks left in drawers, many scientists regard the spontaneous knotting as an urban myth originating in a selective memory for when things go wrong. However, with both tumbling toast5 and odd socks6, I have shown that so dismissive an attitude is ill-placed: both these "urban myths" have entirely rational - and surprisingly deep - explanations, in rigid body dynamics and combinatorics respectively. In what follows, I show that the phenomenon of spontaneous knotting is similarly well-founded. Its roots lie in a recently-discovered theorem in topology, found during attempts to solve a puzzle in polymer chemistry.
SELF-AVOIDING RANDOM WALKS
The behaviour of linear polymer chains has long been the focus of research ranging from the flow properties of plastics to the transcription of genetic information on that most famous of polymer molecules, DNA. In the early 1960s, a number of researchers pointed out that as these linear chains grow in length, they appear more likely to become tangled up with themselves. If the two ends of the chain were then bonded through some chemical reaction, these entanglements then became permanently locked within the resulting ring. This phenomenon became known as the Frisch-Wasserman-Delbruck ("FWD") conjecture7,8: the probability of a ring polymer containing at least one knot tends to unity with increasing ring length (Frisch & Wasserman 1961; Delbruck 1962).
This conjecture was finally proved in 1988 by Sumners and Whittington using the mathematics of topology9. Their proof is based on the idea that a tangled polymer chain can be modelled as a random "self-avoiding walk", that is, a directed random sequence of edges along the simple cubic lattice such that adjacent pairs of edges in the sequences share a common vertex of the lattice but no vertex is visited more than once. Although apparently somewhat abstract, the concept of the random self-avoiding walk (SAW) is eminently applicable to real-life phenomena. For example, it allows the contortions of a polymer chain to be modelled mathematically in a way that captures the fact that no two parts of the chain can occupy precisely the same position - as must be the case for any physical chain of finite thickness. Sumners and Whittington succeeded in proving more than just the FWD conjecture, which is strictly applicable only to closed polygonal structures. They also proved a more general conjecture of direct relevance to the question of spontaneous knotting in ropes: that all except exponentially few sufficiently long SAWs on the simple cubic lattice contain a knotted arc. Both proofs begin by identifying an SAW along the cubic lattice which forms a simple trefoil knot. Let {i, j, k} denote the three orthogonal unit vectors of the lattice in R3. It is then easily seen that, starting at the origin, the following 18-step walk, designated T, constitutes a trefoil knot
T: {i, i, j, k, k, -j, -j, -k, -i, -k, -k, j, j, k, k, -j, i, i} (1)
Using a theorem due to Kesten10 to set a limit on the number of n-step SAWs on which T cannot occur, Sumners and Whittington proved the FWD conjecture concerning polygons, and extended their proof to show that the probability of a SAW containing at least one copy of T is
Prob(1 or more T) > 1 - exp( - k.n + o(n)) k > 0 (2)
where o(n) denotes a rate of increase strictly less than n as n tends to infinity, n being the number of steps of the SAW. This latter result has an obvious interpretation in everyday terms: if a polymer, rope, string, flex or thread is regarded as a SAW, then all except exponentially few sufficiently long ropes etc will contain at least one knot. This certainly appears to constitute a mathematical underpinning of the phenomenon of "spontaneous" knotting. The question is: can the identification of these everday objects with SAWs really be justified? As already noted, the self-avoiding characteristic of SAWs accurately reflects the finite physical extent of knottable objects such as rope. For the knotting process to work, however, the rope must also undergo a random walk process. As long as the lateral stiffness per unit weight of the rope etc. is not too great - thus giving it sufficient flexibility to behave as a SAW - the processes of neglect would indeed seem capable of meeting the random walk requirement. Dumped in a heap in the corner of a garden shed, say, the rope or flex will be thrown around from time to time, allowing it to explore the three-dimensional space in which it is embedded. As the Sumners-Whittington theorem shows, such exploration has an exponentially high probability of including the trefoil sequence (1), which by (2) leads to a high probability of knot creation. Not a great deal of "neglect" is needed to meet this requirement, as is shown by putting some light thread or jewellery chain into a bag and briefly jumbling it up. Carefully extracting the result - to prevent gravity imposing order on the jumbled thread - and pulling on the two ends usually reveals a simple trefoil knot. Indeed, as the length of thread increases, this jumbling manoeuvre proves a remarkably effective way of "forming a knot without really trying". We conclude that neglected rope is indeed adequately modelled by an SAW. As such, the Sumners-Whittington theorem (2) provides a mathematical underpinning of Murphy's Law of Knots.
CONSEQUENCES OF MURPHY'S LAW
An obvious solution to the problem of spontaneous knotting is to coil rope up neatly, and bind it to prevent it performing any random walks within its three-dimensional space. This is, of course, the time-honoured approach of mariners and climbers, who take great care over the storage of ropes. The strong dependence of knotting probability on length suggests another solution, however: divide long lengths in half, and re-join only when needed. Unfortunately, a mathematical quirk of the knotting probability (2) means that Murphy's Law is not so easily circumvented. Consider a long length of, say, electric cable modelled as a self-avoiding walk of 2n steps. Then the probability that it is free of knots is, by (2)
Prob(0 knots in length 2n) = exp( - 2n.k + o(2n)) (3)
Now, to reduce the probability of knotting, suppose we insert a re-connectable break in the cable at its midpoint. The probability that either of the resulting half-cables are free of knots is, again by (2)
Prob(0 knots in length n) = exp( - n.k + o(n)) (4)
which is indeed higher than before. However, as the neglect-driven formation of knots in each half-cable is independent, the probability that both half-cables are free of knots is [Prob(0 knots in length n)]^2. Thus on concatenating the two half-cables, and assuming that this process typically leaves the number of knots unchanged, we have
Prob(0 knots | n joined to n) = exp( - n.k + o(n)). exp (- n.k + o(n)) = exp( - 2n.k + o(2n)) (5)
which is identical to (3). Thus, although each half-cable does have a lower probability of knotting than the length 2n original, when one connects up the two halves again, there is just as much chance of finding at least one knot in the result as in the original uncut cable. The same depressing conclusion is true for the complexity of the knots formed in the cable. Recent research on entanglement complexity of graphs in three-dimensional integer space11 suggests that knot complexity - as characterised by, for example, crossing number - increases at least linearly with length n, and is essentially additive for composite knots. Thus the complexity of a knot formed by concatenating two half-cables each of length n will be at least 2n - just as if the cable had never been divided in the first place. It seems we must abandon our attempts to beat Murphy's Law of Knots by mathematical cunning, and simply store rope, flex and the like as carefully as possible.
CONCLUSION
We have shown that the familiar tendency of neglected rope, flex, string and the like to acquire knots can be accounted for by the properties of self-avoiding random walks. The process of neglect leads to their behaving like SAWs, leading to the "spontaneous" formation of knots whose complexity increases at least linearly with the length of rope. We conclude that the mathematics of topology provides yet further evidence for the reality of Murphy's Law.
ACKNOWLEDGEMENTS
I am delighted to acknowledge the generous interest, advice and encouragement of Professor De Witt Sumners at Florida State University, and Professor Stu Whittington of the University of Toronto. I also thank Geoffrey Budworth for advice on the dangers of random knots, and an anonymous referee for valuable comments.
References
1. Warner, C. & Bednarik, R.G. (1996) Pleistocene Knotting, in History and Science of Knots edited by Turner, J.C., and van de Griend, P. World Scientific (Singapore) 3 - 18
2. Warner, C. (1996) Studies of the behaviour of knots, ibid. 181-203
3. Warner, C. (1996) A history of life support knots, ibid. 149 - 178
4. Budworth, G. (1996) Private communication
5. Matthews, R.A.J (1995) Tumbling Toast, Murphy's Law, and the Fundamental Constants European Journal of Physics 16 172-176
6. Matthews, R.A.J. (1996) Odd Socks: a combinatoric example of Murphy's Law Mathematics Today March-April 39-41
7. Frisch, H. L. Wasserman, E. J. (1961) J. Am. Chem. Soc. 83 3789-3795
8. Delbruck, M. (1962) In Mathematical Problems in the Biological Sciences Proc. Symp. Appl. Math. 14 55
9. Sumners, D.W., Whittington, S.G. (1988) J. Phys. A 21 1689-1694
10.Kesten, H. (1963) J. Math. Phys. 4 960-969
11.Soteros, C. E., Sumners, D.W., Whittington, S.G. (1992) Math. Proc. Camb. Phil. Soc. 111 75 - 91