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What is definition of branched covering?

At the top of page ten of this paper the authors write "the standard two-sheeted branched covering of the sphere by the torus, branched over four points which become the four boundary circles of ...
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Positive vs negative Dehn twist monoids

A general discussion may help clarify these points. There's a lot more to say (e.g. how the positive monoid gives Weinstein fillings, or how negative Dehn twists explicitly give overtwisted disks), ...
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