In general, the tightening of a knot results in changes of its writhe. As the conformation of the knot tends to its most tight, hopefully ideal, conformation, its writhe value tends to a well defined limit. The picture below shows an example.

Fig. 1. Evolution of a loose conformation of the 10.120 knot towards its most tight conformation. The evolution was enforced by  SONO algorithm. Calculations started with a number of segments N=91. At the end of the tightening process the number of segments was doubled 3 times up to N=712.

Numerical experiments on all alternating simple knots with up to 10 crossings reveal, that the limit writhe values show a visible tendency to gather around equally spaced levels. As conjectured by Stasiak and Cerf, the spacing of the levels should be equal 4/7 [2]. Figure below confronts the theoretical prediction with results of the numerical experiments.

Fig. 2.  Wr/(4/7) versus the knot number for ideal conformations of all alternating prime knots with up to 10 crossings. Horizontal lines indicate the writhe levels suggested by Cerf and Stasiak. The gap visible in the set of plotted points in the vicinity of the knot number=80 corresponds to the non-alternating knots with 9 crossings. A smaller gap localized around knot number=40 corresponds to 3 non-alternating knots with 8 crossings.

Is the 4/7 spacing between the writhe levels the optimal one? To answer the question we checked all values from 0 to 1. Figure below shows the results of this check.

Fig. 3. The average relative deviation <|dW|> of the writhe values of all alternating knots with up to 10 crossings from n*qW writhe levels versus the writhe quantum qW.

As seen in the figure above, the 4/7 quantum fits best the numerical data. However, it as also clearly seen from the figure, the fit is not perfect: the writhe values deviate from the Stasiak-Cerf  4n/7 levels. The quantitative analysis of the deviations is presented in the figure below.

Fig. 4. The  probability P of finding a the dW deviations within 20 counting bins. The Gaussian function which fits the data is also plotted; its half width equals 0.2.

The half width of the distribution of the writhe deviations from the closest Cerf-Stasiak levels was shown to be equal 0.2. In view of the analysis of numerical errors we performed, deviations of such a magnitude cannot be attributed to the inaccuracy of the writhe calculations; they must be seen as the intrinsic property of the most tight conformations found by the SONO algorithm. Will a different algorithm of the determination of the ideal conformations substantially reduce the value? Will it reduce it to zero? Is the writhe quasi-quantization an approximate or an exact rule? The questions posed above remain open.

Which is the possible practical implication of the writhe quasi-quantisation phenomenon? To get an insight into the problem, imagine that you are trying to tie different knots, in their most tight forms, on a rope with an internal, parallel threads bundle structure. If at the end of the tying procedure splicing the ends of the rope one wants, for some reasons, to keep identity of the threads, the angle of the relative orientation of the meeting ends should be equal to a multiple of 2p. This happens when the writhe value of the knot is integer. From such a point of view, the tight conformations of the knots whose writhe values are grouped around a non integer writhe level are more difficult to tie than the tight conformations of knots whose writhe value stays close to an integer level. Have a look at the picture below in which two knots with a non integer and an integer values of writhe are presented.

Fig. 5. The ideal 3.1 (left) and 4.1 (right) knots tied on a rope with an internal structure. The parallel transport of the internal structure of the rope leads in the case of the 3.1 knot to a distinct misfit of the orientation of the meeting ends. The ends fit perfectly well in the case of the 4.1 knot – the arrow indicates the hardly visible meeting point.

[1]  P. Pieranski, in Ideal Knots, eds. A. Stasiak, V. Katritch and L. H. Kauffmanm, World Scientific, Singapore, 1998.
[2]  C. Cerf and A. Stasiak, PNAS  97, 3795 (2000)
[3]  P. Pieranski and S. Przybyl, Eur.Phys. J. E 6, 117 (2001).
[4]  P. Hoidn, R. Kusner and A. Stasiak, New J. Phys. 4,  20 (2002).

A pop-science article by Keith Devlin describing the writhe quantization phenomenon can be found in New Scientist.