# Characterizing algebraic tangle by their double branched covers

Montesinos proved that the double branched cover $$\Sigma(T)$$ of an algebraic tangle $$T$$ in a $$3$$-ball is a graph manifold. I wonder if the converse true: Is $$T$$ algebraic if $$\Sigma(T)$$ is a graph manifold?

If unknown, how about this simpler question: Is $$T$$ algebraic if $$\Sigma(T)$$ is a Seifert manifold?

Yes, this is true. Suppose that $$T$$ is the given tangle in the three-ball, and the branched double cover is a graph manifold. By the uniqueness of the JSJ decomposition, the decomposition of the graph manifold into Seifert manifolds descends to give a tangle decomposition of $$T$$. This reduces full case to your simpler case where the branched double cover is a Seifert fibered space with boundary.