"Band sum" isn't a widely recognized term in mathematics or related fields, as of my last update in October 2023. However, the term could possibly be used in various contexts, such as in statistics, computing, or even in specific branches of applied mathematics. If you’ve encountered "band sum" in a particular context, please provide more details.
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A band sum joins two embedded spheres or links by a narrow tube along an arc, removing a small disk from each and joining their boundaries. In a handle slide, the construction uses a parallel framed copy of the second attaching sphere and transports its framing of an embedded sphere along the band.