DNA and other polymer chains in confined spaces behave like closed loops. Arsuaga et al. \cite{AB} introduced the uniform random polygon model in order to better understand such loops in confined spaces using probabilistic and knot theoretical techniques, giving some classification on the mean squared linking number of such loops. Flapan and Kozai \cite{flapan2016linking} extended these techniques to find the mean sum of squared linking numbers for random linear embeddings of complete graphs $K_n$ and found it to have order $\Theta(n(n!))$. We further these ideas by inspecting random piecewise-linear embeddings of complete graphs and give introductory-level summaries of the ideas throughout. In particular, we give a model of random piecewise-linear embeddings of complete graphs where the number of line segments between vertices is given by a random variable. We find further that in our model of the random piecewise-linear embeddings, the order of the expected sum of squared linking numbers is still $\Theta(n (n!))$.

Author Bio

Yasmin is currently a senior at Swarthmore College, who is majoring in mathematics and minoring in music. Outside of her interest in mathematics, she is an avid oboist, and she enjoys learning about dance and reading about psychology in her spare time. In the future, she plans to pursue a doctorate in mathematics.

Xingyu is currently a first year graduate student in mathematics at University of North Carolina at Chapel Hill. He was an undergraduate rising senior at Purdue University during the period in which this work was completed. He is broadly interested in geometry and topology.

Spencer is a senior undergraduate at the University of Kentucky and Math Club Co-President. He is planning to apply to graduate school for mathematics. He has a strong affinity for all things mathematics and enjoys looking for new math problems in my everyday life.

Pedro was a mathematics major at the University of Maryland, College Park. Besides mathematics, he loves reading Latin-American literature and learning about linguistics.