Given a knot $K \subset S^3$ and a rational number $p/q \in \mathbb{Q}\cup\{\infty\}$ (the surgery coefficient), the Dehn surgery $S^3_{p/q}(K)$ is constructed as follows: (1) Remove the tubular neighborhood $N(K)$ to get the knot exterior $E_K = S^3\setminus\mathring{N}(K)$, whose boundary is a torus $T^2$. (2) Reglue a solid torus $D^2\times S^1$ by a homeomorphism $\phi: \partial(D^2\times S^1) \to \partial E_K$ that sends the meridian $\partial D^2 \times\{*\}$ to the curve $p\mu + q\lambda$ on $\partial E_K$. The result is a closed 3-manifold. The case $p/q = \infty$ (or $1/0$) recovers $S^3$.

(Lickorish-Wallace Theorem) Every closed, connected, orientable 3-manifold can be obtained by integer Dehn surgery on a framed link $L \subset S^3$. That is, for any such manifold $M$, there exists a link $L = K_1\cup\cdots\cup K_n$ and integers $p_1,\ldots,p_n$ such that $M = S^3_{p_1}(K_1)\cup\cdots$ (simultaneous surgery). Moreover, the surgery link can be chosen to be a framed link of unknotted components (Lickorish's strengthening), requiring at most $3g$ components where $g$ is the Heegaard genus of $M$.

The fundamental group of $S^3_{p/q}(K)$ is obtained from the knot group by adding one relation: $\pi_1(S^3_{p/q}(K)) = \pi_K / \langle\mu^p\lambda^q\rangle^N$ where $\langle\mu^p\lambda^q\rangle^N$ denotes the normal closure. Integer surgery ($q = 1$): $\pi_1(S^3_p(K)) = \pi_K/\langle\mu^p\lambda\rangle^N$. For the unknot: $\pi_1(L(p,1)) = \mathbb{Z}/\langle p\rangle = \mathbb{Z}_p$. The first homology $H_1(S^3_{p/q}(K)) = \pi_1^{\mathrm{ab}} = \mathbb{Z}/p\mathbb{Z}$ for $p \neq 0$ (independent of the knot for $q = 1$).