Connected sum of two cylinders

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What is the connected sum of 2 cylinders (polygon with identifies edges) if we are doing via atlas Whenever I am adding, I am left with 3 holes instead of 4 and I don't know how to explain here without figure

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-->-- -->-- | | + | | -->-- -->-- If these are 2 cylinders with 2 holes each, after taking connected sum the resultant gives 3 holes only