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In
geometric topology In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originat ...
, a band sum of two ''n''-dimensional
knots A knot is a fastening in rope or interwoven lines. Knot may also refer to: Places * Knot, Nancowry, a village in India Archaeology * Knot of Isis (tyet), symbol of welfare/life. * Minoan snake goddess figurines#Sacral knot Arts, entertainmen ...
''K''1 and ''K''2 along an (''n'' + 1)-dimensional 1-handle ''h'' called a ''band'' is an ''n''-dimensional knot ''K'' such that: * There is an (''n'' + 1)-dimensional 1-handle ''h'' connected to (''K''1, ''K''2) embedded in ''S''''n''+2. * There are points p_1\in K_1 and p_2\in K_2 such that h is attached to K_1\sqcup K_2 along p_1\sqcup p_2. ''K'' is the ''n''-dimensional knot obtained by this surgery. A band sum is thus a generalization of the usual
connected sum In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classific ...
of knots.


See also

*
Manifold decomposition In topology, a branch of mathematics, a manifold ''M'' may be decomposed or split by writing ''M'' as a combination of smaller pieces. When doing so, one must specify both what those pieces are and how they are put together to form ''M''. Manifol ...


References

*. *. Topology Differential topology Knot theory Operations on structures {{knottheory-stub