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In order to model entanglements of polymers in a confined region, we considerthe linking numbers and writhes of cycles in random linear embeddings ofcomplete graphs in a cube. Our main results are that for a random linearembedding of $K_n$ in a cube, the mean sum of squared linking numbers and themean sum of squared writhes are of the order of $\theta(n(n!))$. We obtain asimilar result for the mean sum of squared linking numbers in linear embeddingsof graphs on $n$ vertices, such that for any pair of vertices, the probabilitythat they are connected by an edge is $p$. We also obtain experimental resultsabout the distribution of linking numbers for random linear embeddings of thesegraphs. Finally, we estimate the probability of specific linking configurationsoccurring in random linear embeddings of the graphs $K_6$ and $K_{3,3,1}$.

Linking number and writhe in random linear embeddings of graphs