Linking number and writhe in random linear embeddings of graphs

Abstract

In order to model entanglements of polymers in a confined region, we consider the linking numbers and writhes of cycles in random linear embeddings of complete graphs in a cube. Our main results are that for a random linear embedding of $K_n$ in a cube, the mean sum of squared linking numbers and the mean sum of squared writhes are of the order of $\theta (n(n!))$. We obtain a similar result for the mean sum of squared linking numbers in linear embeddings of graphs on n vertices, such that for any pair of vertices, the probability that they are connected by an edge is p. We also obtain experimental results about the distribution of linking numbers for random linear embeddings of these graphs. Finally, we estimate the probability of specific linking configurations occurring in random linear embeddings of the graphs $K_6$ and $K_{3,3,1}$.

Appendix: Computing q

In order to obtain better estimates of the linking probabilities for $K_6$ and $K_{3,3,1}$, we numerically estimated the value of q. Consider two triangles described by the consistently oriented edges $l_1,l_2,l_3$ and $l_1',l_2',l_3'$, and let $\epsilon _{ij}$ denote the signed crossing number between $l_i$ and $l_j'$. Then it follows from the proof of Lemma 2.2 in [1] that

$$\begin{aligned} E\left[ \left( \sum _{i,j=1}^3 \epsilon _{ij}\right) ^2\right] = 18q. \end{aligned}$$

But the quantity $\sum _{i,j=1}^3 \epsilon _{ij}$ is precisely twice the linking number of the two triangles, which for a linear embedding is either 0 or $\pm 1$. Hence, $\frac{18q}{4}$ is the probability that a random linear embedding of two disjoint triangles is linked.

We wrote a Python program to generate random linear embeddings of two triangles by taking six random points in $C^3$, described as three (pseudo)random coordinates in (0, 1), and computing the associated linking number. Out of the one billion samples generated, 152,402,780 were linked, giving a 99 % confidence interval for q of $0.033867 \pm 0.000013$.

All code is available at http://math.berkeley.edu/~kozai/random_graphs/.