### - Art Gallery -

In geometric topology, a band sum of two n-dimensional knots K1 and K2 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 (K1, K2) embedded in Sn+2.
There are points $${\displaystyle p_{1}\in K_{1}}$$ and $${\displaystyle p_{2}\in K_{2}}$$ such that h is attached to $${\displaystyle K_{1}\sqcup K_{2}}$$ along $${\displaystyle p_{1}\sqcup p_{2}}.$$

K is the n-dimensional knot obtained by this surgery.

A band sum is thus a generalization of the usual connected sum of knots.

Manifold decomposition

References

Cromwell, Peter R. (2004), Knots and Links, Cambridge University Press, p. 90, ISBN 9780521548311.
Kawauchi, Akio (1996), Survey on Knot Theory, Springer, p. 31, ISBN 9783764351243.

Mathematics Encyclopedia

World

Index