Sharpness of the winding-versus-crossing bound

Determine whether the bound $n(\gamma)\leq\min(N_x,N_y)-1$ for the winding number of a bi-rotational self-shrinker profile curve is attained.

Background

Theorem C establishes that, whenever the global winding number is defined, the number of full turns of the profile curve is bounded above by one less than the smaller of its numbers of crossings with the two cylindrical separatrices. The estimate applies to closed orbits and to trajectories whose tangent angle has limits at both ends. For recurrent trajectories, only a local version is available, and the paper explicitly leaves unresolved whether the global inequality can be sharp.