Combinatorics of $k$-Farey graphs

Source abstract

With an eye towards studying curve systems on low-complexity surfaces, we introduce and analyze the k-Farey graphs ℱk and ℱ≤k, two natural variants of the Farey graph ℱ in which we relax the edge condition to indicate intersection number =k or ≤k, respectively. The former, ℱk, 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 ℱk is a quasitree (in fact, a tree when k is even) and Aut(ℱk) is uncountable for k>1. As for ℱ≤k, Agol obtained an upper bound of 1+ min{p:p is a prime>k} 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(mod12), and we show computer-assisted computations of many values of k for which equality fails.