Band sum

Method of connecting knots

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 ${\displaystyle h}$ 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.

See also

• 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.