Remove links by Kirby moves

Question

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?

Answer 1

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.

For the (standard picture of the) trefoil, changing any of the crossings gives you the unknot. For a general knot, you need to find a set of suitable crossing changes. One way to do this is to choose a 'starting point' on the knot and a direction. Then arrange that as you traverse the knot, the first time you arrive at each crossing you get the overcrossing. This will give you the unknot.