Curves and laminations on the five-times punctured sphere

We study laminations on the five-times punctured sphere Σ0;5. The discussion is divided into two parts. The results obtained in the two parts are not directly linked. In particular, each part can be read independently. Firstly, we analyse the inclusion of the curve graph of Σ0;5 into PML(Σ0;5). We completely characterise the image of the induced subgraph on a pentagon and the associated decagon. This enables us to describe explicit loops in the curve graph whose images in PML(Σ0;5) form a Hopf link and a trefoil knot. Secondly, we investigate the topology of superconvergence Ts on the set of boundary laminations BL. Here a boundary lamination is a minimal lamination on Σ0;5 with more than one leaf. We show that (BL;Ts) is path connected. Gabai proved that the space of ending laminations is locally path connected [Gab1]. Using Gabai's theorem, we prove that (BL;Ts) is locally path connected. Combined with work of Brock and Masur [BrocM], this implies that the Gromov boundary of the pants graph of _0;5 is path connected and locally path connected