Visualize the lens space $L(1,1)\cong S^3$

Question

I know $L(p,q)\cong L(p,q-np)$ for all $n\in \mathbb{Z}$, so in particular $L(1,1)\cong L(1,0)\cong S^3$. It is easy to visualize $L(1,0)\cong S^3$ since the complement of an unknotted solid torus in $S^3$ is also a solid torus, and $L(1,0)$ is obtained by glueing the meridian of the fomer one with the longitude of the later one. My question is how to visualize $L(1,1)\cong S^3$. More generally, how to visualize $L(1,q)\cong S^3$?