6
Linear Behavior of the Writhe Versus
the Number of Crossings in Rational Knots
and Links
C. Cerf and A. Stasiak
Summary.Using the formula introduced in [Proc. Natl Acad. Sci. USA 97 , 3795
(2000)], we can predict the 3D writhe of any rational knot or link in its ideal config-
uration, or equivalently, the ensemble average of the 3D writhe of random configura-
tions of it. Here we present a method that allows us to express the writhe as a linear
function of the minimal crossing number within individual Conway families of ratio-
nal knots and links. We discuss the cases of families with slopes± 4 / 7 ,± 10 / 7 ,±1,
and 0. For families with the same slope value, the vertical shift between the corre-
sponding lines can also be computed.
6.1 Introduction
Quantization of writhe in knots is a puzzling phenomenon, which was initially
discussed only among a narrow group of specialists but recently became quite
famous [1]. Let us explain what this concept covers. Writhe (or 3D writhe,
or Wr) is a measure of chirality of oriented closed curves in 3D space. It
corresponds to the average signed number of perceived self-crossings in an
oriented curve when observed from a random direction, where each right-
handed crossing is scored as +1 and each left-handed crossing is scored as−1.
Writhe is usually calculated using a Gauss integral formula [2].
Studies of random walks in a cubic lattice revealed that, while different
realizations of random knots of a given type have stochastically distributed
values of their writhe, the average of writhe over the statistical ensemble
of knots of a given type, like right-handed trefoils for example, reaches a
characteristic value that is independent of the length of the walk [3, 4]. Thus
for example the average writhe of random right-handed trefoils in a cubic
lattice does not change when the number of segments is increased from 34
to 250. The same studies pointed out that there is a constant increase (i.e.,
aquantization) of the average writhe between successive knots belonging to
families of torus knots like 3 1 , 51 , 71 , 91 , etc. or to twist knots like 4 1 , 61 , 81 ,etc.
The specific difference of average writhe between successive knots belonging
to different families depends on the particular family of knots.