Annular and pants thrackles

A thrackle is a drawing of a graph in which each pair of edges meets precisely once. Conway's Thrackle Conjecture asserts that a thrackle drawing of a graph on the plane cannot have more edges than vertices. We prove the Conjecture for thrackle drawings all of whose vertices lie on the boundaries of d ≤ 3 connected domains in the complement of the drawing. We also give a detailed description of thrackle drawings corresponding to the cases when d = 2 (annular thrackles) and d = 3 (pants thrackles).