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Every energy level of a charged quantum particle confined in a regionthreaded by a magnetic flux line with quantum flux one-half must be degeneratefor some position of the semifluxon within the boundary B. This is illustratedby computations for which B is a circle and a conformal transformation of acircle without symmetry. As the shape of B is varied, two degeneracies betweenthe same pair of levels can collide and annihilate. Degeneracy of three levelsrequires three shape parameters, or the positions of three semifluxons;degeneracy of N levels can be generated by int{N(N+1)/4} semifluxons. The forceon the semifluxon is derived.

Semifluxon degeneracy choreography in Aharonov-Bohm billiards