An infinite cyclic cover for the complement of a knot $K$ can be constructed by performing a suitable Dehn surgery that unknots $K$.
For example, in the following picture we take $6_1$ and use Dehn surgery to eliminate the twist. The result is unknotted, so the infinite cyclic cover is easy to describe as stacked cylinders. By keeping track of the blue curve and the red meridian, we can recover the original knot by another Dehn surgery. In particular, the infinite cyclic cover of $6_1$ is the result of $-5/2$ surgery on all of the blue curves simultaneously.
After the first step, the red curve shows that $-1/2$ surgery recovers $6_1$. In the infinite cyclic cover of the unknot, this becomes $-5/2$ surgery on the curves to recover the infinite cyclic cover of $6_1$.
Without drawing the red curves, how can one compute $-5/2$ from $-1/2$ and the blue knot? What is a good reference to learn how to do these computations?