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You can deform the embedding into an homotopically distinct embedding with the same image, thus changing the homotopy classes of curves that correspond to blue regions. Indeed, you can deform your embedding to a more symmetric one: two concentric spheres, one slightly inside the other, with three "collars" between them (each collar being obtained by removing two disks, one in each sphere, and replacing them with an annulus), regularly positioned. Then by a one third of turn rotation and the inverse deformation, you obtain an embedding with the same image as before, but with one of the previously blue curve sent to the outer most curve.