# Remove links by Kirby moves

I am trying to prove the following proposition.

proposition; If in a framed link $L$ a component $K$ is an unknot with framing zero which links only one other component $H$ geometrically once, then $K \cup H$ may be moved away from the link $L$ without changing framings and cancelled.

This is Proposition 3.3 of the book "Lectures on the topology of 3 manifold" by Nikoli Saveliev. I understood the first half of the proof.

What I don't understand is to show that we can get rid of $K \cup H$. The proof of the book says;

The same move changes crossings of $H$ itself, thereby unknotting $H$ and changing its framing by an even integer. We ended up with the link in Fig 3.22...

Here the same move means that we do a Kirby move with $K$ (sliding a components over $K$). Figure 3.22 is the Hopf link with framing $0$ and some integer $p$.

Could you tell me how to change the knot $H$ to an unknot? (and then we have a Hopf link with that unknot and $K$.)

I tried to do this for the case when $H$ is trefoil but it seems complicated. Is there any algorithm to do this? Or do we prove this just abstractly?

You can convert $H$ to an unknot by changing crossings. The crossing changes are effected by sliding over $K$ -- by an isotopy you can draw $H$ close to a crossing as a meridian of the strand crossing over and the slide the strand of the undercrossing over it.