Connected Sum

The connected sum $M_1\char93 M_2$ of $n$-manifolds $M_1$ and $M_2$ is formed by deleting the interiors of $n$-Balls $B_i^n$ in $M_i^n$ and attaching the resulting punctured Manifolds $M_i-\dot B_i$ to each other by a Homeomorphism $h:\partial B_2\to \partial B_1$, so

$$M_1\char93 M_2 = (M_1-\dot B_1)\bigcup_h (M_2-\dot B_2).$$

$B_i$ is required to be interior to $M_i$ and $\partial B_i$ bicollared in $M_i$ to ensure that the connected sum is a Manifold.

The connected sum of two Knots is called a Knot Sum.

See also Knot Sum

References

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 39, 1976.