Fundamental theorem of Dehn surgery

Every closed connected orientable three-manifold is obtained by integer Dehn surgery on a finite framed link in the three-sphere. Compare a Heegaard splitting with a standard splitting of the sphere, factor their gluing difference using the Lickorish-Dehn theorem, and realize each twist by unit surgery on a distinct level of a collar of the splitting surface.

Put a simple closed curve in a surface collar and use its surface framing. Filling its deleted tubular neighborhood along $\mu \pm \lambda$ inserts one full positive or negative twist of the annulus between the collar levels. Thus surgery of coefficient $\pm 1$ changes the gluing by a Dehn twist or its inverse. The exact sign depends on the orientations of the seam and normal direction.