Band sum

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.

ee also

*Manifold decomposition