# Homology torsion in the double branched cover of a tangle?

Let $$T$$ be a locally unknotted $$2$$-tangle in $$B^3$$ and $$\Sigma(T)$$ be its double branched cover. Can $$H_1(\Sigma(T))$$ have a non-trivial torsion? (Obviously, not for rational tangles.)

• I feel like there should be plenty of alternating tangles with this property, such as pretzel or algebraic tangles. – Ian Agol Apr 28 at 4:34

An example (similar to Ken's and related to Ian's comment): take the pretzel tangle P(-3,-3,3) drawn below; its double branched cover has homology $$Z \oplus Z/3 \oplus Z/3$$. This is from my paper (Embedding tangles in links, J. Knot Theory Ramif.,9, No. 4 (2000), 523-530).