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In geometric topology, a band sum of two $n$ -dimensional knots $K 1$ 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 of knots.

[edit] See also

Manifold decomposition