Dehn filling

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 $\alpha$.

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.