#### Article Title

The Mean Sum of Squared Linking Numbers of Random Piecewise-Linear Embeddings of $K_n$

#### Abstract

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 $\Theta(n (n!))$.

#### Faculty Sponsor

Kenji Kozai

#### Recommended Citation

Aguillon, Yasmin; Cheng, Xingyu; Eddins, Spencer; and Morales, Pedro
(2023)
"The Mean Sum of Squared Linking Numbers of Random Piecewise-Linear Embeddings of $K_n$,"
*Rose-Hulman Undergraduate Mathematics Journal*: Vol. 24:
Iss.
2, Article 3.

Available at:
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/3

#### Included in

Discrete Mathematics and Combinatorics Commons, Geometry and Topology Commons, Other Mathematics Commons