Band sum

Short description: 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 [math]\displaystyle{ p_1\in K_1 }[/math] and [math]\displaystyle{ p_2\in K_2 }[/math] such that [math]\displaystyle{ h }[/math] is attached to [math]\displaystyle{ K_1\sqcup K_2 }[/math] along [math]\displaystyle{ p_1\sqcup p_2 }[/math].

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, https://books.google.com/books?id=djvbTNR2dCwC&pg=PA90 .
• Kawauchi, Akio (1996), Survey on Knot Theory, Springer, p. 31, ISBN 9783764351243, https://books.google.com/books?id=gWbyJn7c5G0C&pg=PA31 .