Dehn surgery

Dehn surgery removes tubular neighborhoods of a link in a three-dimensional manifold and performs Dehn filling on their boundary tori with specified primitive meridian slopes. Dehn surgery on a framed link is the integral-slope special case.

Let $X$ be a three-manifold with a torus boundary component and let $\alpha$ be a primitive slope on that torus. The Dehn filling $X(\alpha )$ glues in a solid torus whose meridian is identified with $\alpha$.

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.

Let a knot lie in a fiber of a mapping torus of an oriented topological surface. Surgery with coefficient $-1$ relative to its surface framing replaces the monodromy by its composition with a right-handed Dehn twist. Coefficient $+1$ gives the inverse twist. The composition side depends on the choice of seam and time orientation; the existence of the new fiber bundle does not.

For a null-homologous knot, choose a meridian of a knot $\mu$ and Seifert longitude $\lambda$. Rational surgery with relatively prime numerator $p$ and denominator $q$ fills the knot exterior along $p\mu +q\lambda$. A negative continued fraction for $p/q$ replaces this operation by integer surgery on a chain of meridians, using slam-dunk moves.

Let two parallel curves ${\beta }_1,{\beta }_2$ cut a torus boundary into annuli $A_1,A_2$, and let a slope $\alpha$ meet each ${\beta }_i$ once. Identifying $A_1$ with $A_2$ by an orientation-reversing map fixing their boundary and matching the two arcs of $\alpha$ produces the same manifold as the Dehn filling $X(\alpha )$. A collar of the boundary turns the quotient region into the solid torus attached by the filling.