Question

Hello,

If you have a link of 2 components $U$ and $V$ in $S^3$. $U$ is the unknot, and you make $1/q$ dehn filling in $U$, you can visualize the resulting knot $V'$ from $V$, by twisting it $q$ times along a disc bounding $U$. How can I visualize $p/q$ dehn filling?. I mean, in that case after you do the filling you end up with a knot in the lens space $L(p,q)$, what knot $V'$ do you get in the universal covering $S^3$?

Thanks!

Answer 1

The complement of $U$ is a copy of $S^1\times D^2$. Let $\widetilde V$ be the $p$-fold cover of $V$ in the $p$-fold cover of $S^1\times D^2$. Cut $\widetilde V$ along $pt \times D^2$, twist $q$ times, then reglue to get a link $\widetilde V_t$ in $S^1\times D^2$. Embed the modified $S^1\times D^2$ back into $S^3$ in the obvious way. The resulting image of $\widetilde V_t$ is what you are after.

Note that $\widetilde V_t$ might have multiple components (if the linking number of $V$ with $U$ is not a multiple of $p$), so in general it will be a link, not a knot.