Cyclic Graphs and Winding Fraction

Main Theorem

A major result of previous work is that the homotopy type of the clique complex of any cyclic graph depends only on its winding fraction. These details can be understood in the following informal manner:

Definition. A directed graph $$G$$ is cyclic if its vertices can be placed in a cyclic order such that, whenever there is an edge $$u \to w$$, then there are also edges $$u \to v \to w$$ for all $$u \prec v \prec w$$.

Definition. For a cyclic graph $$G$$ and a vertex $$v$$, define $$f(v)$$ to be the clockwise-most vertex $$w$$ such that there exists an edge $$v\to w$$. Then the winding fraction of $$G$$ is

$\mbox{wf}(G) = \sup\bigg\{\frac{\omega}{k}\bigg\}$ where the ratio $$\omega/k$$ is realized by an $$f$$-periodic orbit with length $$k$$ which "winds" $$\omega$$ times around its center, and the $$\sup$$ is taken over all $$f$$-periodic orbits in $$G$$.

Definition. A cyclic graph is called metric if its vertex set can be given a metric $$d$$ such that $$u \prec v \prec w$$ implies $$d(u,v)< d(u,w)$$.

For technical reasons, metric cyclic graphs behave much more nicely than mere cyclic graphs. Indeed, the existence of the nearly monotone metric guarantees that certain $$f$$-periodic orbits unique, many functions from $$G$$ into $$\mathbb{R}$$ are continuous, etc. While we omit these details for brevity's sake, a main contribution of this project was the development of this general theory of metric cyclic graphs.

All this in mind, we are able to reduce this topological task to a purely geometric one: When are (metric) cyclic graphs supported in $$P_n$$ and how does the winding fraction change over these regimes?