Dehn surgery

Remarks

For a knot $K$ in $S^3$, let $\mu$ be a meridian, bounding a disc in $N(K)$, and $\lambda$ a longitude that is null-homologous in $E$, both oriented. Each slope is isotopic to a curve representing $p\mu + q\lambda$ in $H_1(\partial E; \Z)$ with $p, q$ coprime, and the slope is labelled by $p/q \in \Q \cup \set{\infty}$. The slope $\infty = 1/0$ gives back $S^3$; a slope with $q = \pm 1$ is integral.

Example: let $U$ be the unknot in $S^3$. Its exterior is a solid torus in which $\lambda$ bounds a disc. Surgery with slope $0$, that is $\alpha = \lambda$, glues two solid tori so that their meridians match, and the result is $S^2 \times S^1$. Surgery on $U$ with slope $p/q$, $p \ne 0$, gives a lens space $L(p, q)$, up to the orientation convention.

Non-example: a curve $\alpha$ that bounds a disc in $\partial E$ is not a slope. No homeomorphism $h$ can send $\partial D^2 \times \set{1}$ to it, because a homeomorphism of tori sends essential curves to essential curves.