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Can a black hole be presented as a Heegaard decomposition or as the complement of a knot?

I'll try and elaborate: If I understand correctly, the cross section of spacetime near a black hole can be thought of topologically as a manifold. What manifold is it? Can the manifold be decomposed?

I'll try and elaborate: If I understand correctly, the cross section of spacetime near a black hole can be thought of topologically as a manifold. What manifold is it? Can the manifold be decomposed?

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