Combinatorics of $k$-Farey graphs

Abstract

With an eye towards studying curve systems on low-complexity surfaces, we introduce and analyze the $k$-Farey graphs ${\mathcal{ℱ}}_k$ and ${\mathcal{ℱ}}_{\le k}$, two natural variants of the Farey graph $\mathcal{ℱ}$ in which we relax the edge condition to indicate intersection number $=k$ or $\le k$, respectively.

The former, ${\mathcal{ℱ}}_k$, is disconnected when $k>1$. In fact, we find that the number of connected components is infinite if and only if $k$ is not a prime power. Moreover, we find that each component of ${\mathcal{ℱ}}_k$ is a quasitree (in fact, a tree when $k$ is even) and $Aut\left({\mathcal{ℱ}}_k\right)$ is uncountable for $k>1$.

As for ${\mathcal{ℱ}}_{\le k}$, Agol obtained an upper bound of $1+min\left\{p:p\text{is a prime}>k\right\}$ for both chromatic and clique numbers, and observed that this is an equality when $k$ is either one or two less than a prime. We add to this list the values of $k$ that are three less than a prime equivalent to $11\left(mod12\right)$, and we show computer-assisted computations of many values of $k$ for which equality fails.