There are exactly seven critical complexes of the form (G × S¹) ∪ H that are minimal obstructions to embedding in S³: two K₄-type, four Θ₄-type, and one K_{2,3}-type.
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A Kuratowski-Type Classification of Critical Complexes for the 3-Sphere
There are exactly seven critical complexes of the form (G × S¹) ∪ H that are minimal obstructions to embedding in S³: two K₄-type, four Θ₄-type, and one K_{2,3}-type.