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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?

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    $\begingroup$ Please note that most of us won't have a copy of the book in front of us, so you need to explain all the notation that you use. For instance, what are $F$, $S_+$ and $B_+$? $\endgroup$
    – HJRW
    Commented Apr 4, 2021 at 14:35
  • $\begingroup$ This is a technical lemma used for a proof that a link which admits a connected alternating diagram has an irreducible complement. Both $F$ and $S^+$ are spheres, and $B^+$ is a ball bounded by $S^+$. But I agree, the question should provide much more background information. $\endgroup$
    – Josh Howie
    Commented Apr 4, 2021 at 21:23

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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.

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  • $\begingroup$ Thanks a lot! I am still confused about the saddles as their boundary intersects in four points that divide it into four arcs; Is there any such a preferred position of the saddles as they should be around the equator of the bubbles? $\endgroup$
    – user174967
    Commented Apr 4, 2021 at 22:12
  • $\begingroup$ Take a look at figure 4 in the linked article of Menasco. This is the position of a saddle. If there are multiple saddles in a bubble, then they are parallel copies of one saddle. $\endgroup$
    – Josh Howie
    Commented Apr 4, 2021 at 22:18

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