Answering a question of Wright, we show that spheres of any radius are always connected in the curve graph of surfaces Σ_2,0, Σ_1,3, and Σ_0,6, and the union of two consecutive spheres is always connected for Σ_{0, 5} and Σ_1,2. We also classify the connected components of spheres of radius 2 in the curve graph of Σ_0,5 and Σ_1,2.

All authors are funded by NSF grant DMS-2142712.