As we know, the Lens space $L(p,q)$ is the quotient of $S^3$ by a $\mathbb{Z}_p$ action: $(z_1,z_2) \rightarrow (e^{2\pi i/p}z_1,e^{2\pi iq/p}z_2)$.

It seems that the Lens space $L(1,0)$, a.k.a $S^3$, could be realized by surgery along the Hopf link in $S^2\times S^1$. The Hopf link could be represented by the torus link $T(2,2)$.

So I was wondering if the Lens space $L(p,1)$ could be obtained by surgery along the torus link $T(2,2p)$ in $S^2\times S^1$?

any$T(2,2q)$, and this is a good exercise with handle diagrams. $\endgroup$ – Marco Golla Sep 27 '16 at 7:55