A branched surface B in a 3-manifold M is a space locally modeled on Figure 1.5(a), with neighborhood N(B) vertical fiber-shaped of intervals and horizontal and vertical boundary 8hN(B), 8vN(B) as shown in figure 1.5(b). The definition of when a lamination is (entirely) carried by a branching surface is the same as with train tracks. There are definitions of when a lamination (co-dimension one in a 3-manifold) is called essential, and also when a branched surface is called essential.
So here we first define an essential branched surface, and then we define a lamination as essential if it is carried by an essential branched surface. The branched surface is described by first isotopizing it, so that the height function on it is Morse, i.e. Example: Figure 1.10(a) shows a surface (a branched surface without branch points) whose boundary (thick lines) is a trefoil.
As another example, Figure 2.12 shows a branched surface in the trefoil complement similar to the surface in Figure 1.10(a). The latter is actually a subset of the former; it also qualifies as a laminate carried by the ramified surface of Figure 2.12.
Chapter 2
Main Results and Constructions
- Outline
- Construction for Theorem 1
- Construction for Theorem 2
- Chapter 3
In [H] it is shown that this branched surface carries lamellae for all boundary slopes E (-00,0] (Figure 2.13). But there is a problem: if we make only the saddles linear, the boundary curves will not be closed lamination (Figure 2.15) In Figure 2.14, make the saddle in Diagram 1 so that the horizontal lines in Diagram 2 look like Figure 2.16.
We now claim that in Figure 2.15, the saddle in diagram 5 can be chosen such that in diagram 6, the maps on the two vertical lines are mirror images, and thus in the last diagram, both lines can become the identity. Proof of claim: The saddle of diagram 5 of figure 2.15 will be done in several stages (or parts). Then we cut Xl from the right side of the lower line and YI from the right side of the upper line, obtaining figure 2.17 (b).
In figure 2.16 we choose a, Xi and Yi so small that not only does this process converge, but enough of the two horizontal lines will remain that we will get figure 2.18. This is the main difference with the case of the piano, where this hall was only linear (diagram 2 of . figure 2.14). Furthermore, the knot with coefficient r I (r + s) (last diagram) is separate from, and isotopic to, a leaf of the lamination (figure 2.23). Therefore, by [G1) (operation 2.4.2), for all operations except one2 on the unknot, the lamination (on [2,2nD) remains essential.
The purpose of this is to get things working at the transition from diagrams 6 to 7 in Figure 2-25, which is the crucial point of the construction. This proves the claim that the bifurcated surface of Figure 2.25 forms laminations of all boundary slopes. In Diagram 2 of Figure 2-25, before making the saddle, we can obtain any integer slope by turning one of the inner spirals the desired number of times.
And then we can get all (rational) pitches between any two integer pitches by branching the inner spiral and twisting a copy of it, as shown in Figure 2.29.
Proofs of Essentiality
Some Generalities
Next, before splitting T over the collar T x [0,1]', we first take a copy of one side of each digon and move it parallel to itself across the digon to the other side as t goes from ° to 1/2 go. The reason for 'moving copies of sides of digons from one side to the other' is to make analysis of the complementary components of B' CM (and thus checking the essence of B' CM) easier, as explained in the following three lemmas (which, although not in so many words, are proved in theorem 1 of [R]). It is assumed that B satisfies the hypothesis required for construction 3.1.1, namely that 8B is a train track in 8(53 - N(K)) whose complementary areas are all digons.}.
Each annulus consists of a finite union of disks Di ~ I x I with each Di contained alternately in 8vN(B) and 8N(K). The result of 'moving copies of sides of digons across from one side to the other' (in the T x [0,1/2] part of construction 3.1.1) is that we glue one to each complementary component X D2 x I for each Di C 8N (K) in the suture of X. The suture on D2 x I is an annulus, half of which is in 8vN(B'), the other half is glued to Dj.
Therefore, X remains topologically the same, and furthermore its seam remains the same, except that it is now entirely C 8vN (B'). This shows that 8hN(B') has no compression disks or monogons in augmented X, since any such compression disk or monogon could be isotopized. However, it is easily seen that this holds because, by Lemma 3.1.2, all these complementary components are just products of D2 x I, each with a ring oD2 x I as a seam.
Then f3 orR must intersect in at least one point (when entering and exiting R), showing that there are at least two simple closed curves a1 U a2 C Tn or R transverse to the meridians of T. So just as a meridian m crosses A from a1 to a2, in each digon it goes from one side of a D2 x I complementary component of R - C to the other. Each meridional disk of C has two distinct sides: the first side is a full side, say D2 x 0, of a D2 xl, while the second side is a true subset of a D2 x 1 side of a D2 x I, the rest lying on OR.
0:1 at a point on oR, goes to the first side of a meridional disk, emerges from the second side, goes to the first side of perhaps another meridional disk, emerges from the second side, and so on.
Proof of Theorem 1
The complementary component of B trapped between the level 2 spheres of diagrams 1 and 3 is also seen (with a little visualization) to be topologically a D2 x I ball with 8D2 x I as a suture, as shown in Figure 3.38 (where the suture is only drawn as a curve rather than a ring). The complementary component between diagrams 3 and 7 is exactly the same as that between 1 and 3, except that it is upside down. The complementary component between diagrams 7 and 10 is a solid torus A x I with the suture consisting of two parallel rings 8A x I of slope 1 at the boundary of the solid torus, as in Figure 3.39 which is equivalent to Figure 3.36 (again suture drawn as curves).
Both of these cases arise for the non-torus 2-bridge knots, which we deal with in Lemma 2.2.1. For the other torus nodes (and in fact all 2-bridge nodes with the branched surfaces of [H]) we get the same complementary components since we use the same "saddle moves", which yield branched surfaces consisting of the same "building" blocks". There are nice pictures of these building blocks in figure 3.1 of [FH] - for 2-bridge knots only EA, Ee and ED are used.).By cutting and pasting at the branch locus, we can assume that this sphere is embedded in N (B' is).
Then D, together with the horizontal boundary ring aA x IUA xl, plus another copy of D gives a sphere carried by B, which again leads to a contradiction. In Figure 3-39, each curve on the axe, and in particular each Ci, has a connection number of zero to the core of the solid torus (K is a two-bridge knot, so it goes down through the "hole" of the solid torus , and then up, exactly twice). Cm, so aD also has a connection number of zero with the core of the massive torus, which contradicts the fact that aD is homotopic with a meridian.
This part of the proof is different from that given in [H]; there, it is argued that due to a symmetry that the branched surface has, the next seam in the solid torus must also connect a contact disc, thus again yielding a sphere. carried by the branching surface, which gives a contradiction. Here, however, the branching surfaces constructed for the nodes of the torus as in Figure 2.20 no longer have this symmetry—the symmetry exists as long as at most one of the saddles of diagrams 2, 5, or 8 is present—and so we use a different argument.). We showed in the previous chapter that Bo (the branched surface before the enlargement of step 1) fully carries a lamination with closed boundary curves, and the whole argument was essentially qualitative: the main point was to obtain in Figure 2.18 the point free self fixed. -homeomorphisms 9 and h of the interval.
Also in Figure 2.18, the conjugation map h will now be distributed over the saddles of the diagram.
Proof of Theorem 2
But this is also achieved for B simply by assigning small enough weights to all parts of B - Bo, since perturbations 9 and h under sufficiently small perturbations will still satisfy the (open) property of being free from a fixed point to be. It is an unknotted solid torus A x I with one suture ring (the curve in Figure 3-41 is connected). A meridional disk of Ax I should bound a longitude on the massive torus of Figure 3.41, which intersects the suture in 2m + 1 points, m ~ 1; so there are no compression disks or monogons.
We use [G2)'s disk parsing technology (explained below in Section 3.4) to simplify Figure 3.42 to Figure 3.43 by filling in the extra 2p gaps; according to Lemma 3.4.1, one has an essential horizontal boundary if and only if the other has too. But Figure 3-43 is the same as Figure 3-41, so it has no compressive discs or monogons. In fact, to demonstrate the essence of the specific laminations (carried by B') that we have in mind, it suffices to show that B' does not have a contact disk that is isotopic with a sheet of the lamination [G3].
However, we will also show that B' has no contact disks (isotope for a leaf, or not). So just like case 1 of step 4, B degenerates into one of the branched surfaces of [HT], say Bl. Then D is a contact disk for B~, which we know is impossible by the proof of Theorem 1 (or Theorem 1 of [HD.
Essentiality is Preserved by Disc Decomposi- tion
Bibliography
![Figure 2.12: [H]'s branched surface for the trefoil.](https://thumb-ap.123doks.com/thumbv2/123dok/10401514.0/48.864.129.683.99.845/figure-2-12-h-s-branched-surface-trefoil.webp)
![Figure 2.13: [H)'s construction of laminations of slopes (-00,0].](https://thumb-ap.123doks.com/thumbv2/123dok/10401514.0/49.849.90.740.80.783/figure-2-13-h-construction-laminations-slopes-00.webp)
