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In your case, a.touches(A) returns False because documentation says:

touches returns True if the objects have at least one point in common and their interiors do not intersect with any part of the other.

Also A.contains(a) returns False because a lot of points of a lie in the extrior of A. Documentation says:

contains returns True if no points of other lie in the exterior of the object and at least one point of the interior of other lies in the interior of object.

a.difference(A) returns empty geometry because all interior of a is in A. But A.difference(a) gives: