I am reading GTM 175 An introduction to knot theory by Lickorish and have some questions on the proof of Lemma 4.5 given.


For (a), it says "Suppose that $C$ is amongst the $n$ components of $F\cap S\_+$ that do not bound disc components of $F\cap B\_+$."; I was wondering how could $C$, as a component of $F\cap S\_+$, not bounding a disc component, since I am assuming that the intersection of two 2-spheres is always disjoint unions of simple closed curves, and therefore $C$ must bound a disc component in $F\cap B\_+$.


Also, for (b), I wonder if the "saddles" mentioned in this proof are the same as the saddles mentioned before in the right of Figure 4.1. I would like to make sure that, on the bottom of page 34, "...then they are both on opposite sides of the $same$ saddle", it means the over-pass arc, but I was confused why he bother using the term "saddle".
On page 35, it says "Thus $p\_1$ and $p\_2$ are ... on adjacent saddles.". What are the adjacent saddles here? Aren't there only two saddles in this plot?



Questions on the proof Lemma 4.5 GTM 175, Lickorish
For (a), $C$ does bound a disk in $B^+$ which is later called $\Delta'$, but by assumption $C$ does not bound a disk in $F\cap B^+$. A priori, the components of $F\cap B^+$ are subsurfaces of the sphere $F$; they could be disks, annuli, pairs of pants, etc.


For (b), the surface $F$ could intersect a bubble in any number of saddles. These saddles inherit an ordering from the height at which they intersect the axis of the bubble, so it makes sense to talk about adjacent saddles with respect to this ordering. When he talks about them being on the opposite sides of the same saddle, it is because $p\_1$ and $p\_2$ lie on the highest saddle but $p\_1$ and $p\_2$ lie on opposite sides of $L$ in the intersection of $S^+$ with the boundary of this bubble.


For a slightly different proof, which also uses these standard position arguments, you might like to look at the original proof by [Menasco](https://doi.org/10.1016/0040-9383(84)90023-5).


