Persistent laminations from Seifert surfaces

Mark Brittenham^{1} and Rachel Roberts^{2}

University of Texas at Austin

Abstract. We show how an incompressible Seifert surface F for a knot Kin S^{3} can be used to create an essential lamination L_{F} in the complement of each of an infinite class of knots associated to F. This lamination is persistent for these knots; it remains essential under all non-trivial Dehn fillings of the knot complement. This implies a very strong form of Property P for each of these knots.

§0

Introduction

Essential laminations have found many uses in 3-manifold topology, principally in understanding Dehn fillings on 3-manifolds. For example, Delman [De] showed that 2-bridge knots have Property P, by finding essential laminations in their complements which remain essential under all non-trivial Dehn fillings. Such laminations are called persistent. Naimi [Na], independently, proved the same result using different methods. In fact, since every Dehn filled manifold contains an essential lamination, each has universal cover ℝ^{3} [GO], a property which we have chosen to call \underline{strong} Property P for the knot. Since reducible manifolds do not have universal cover ℝ^{3}, strong Property P implies the cabling conjecture for the knot, as well.

Delman and Roberts [DR], using similar methods, have proved strong Property P for non-torus alternating knots. Recently, Wu [Wu] proved strong Property P for most algebraic knots, again using essential laminations.

In this paper we describe a process for generating knots with strong Property P, by building essential laminations in the complement of a knot using an incompressible Seifert surface for some other knot. The knots which we succeed in doing this for are usually not alternating, and are probably not algebraic, since they do not seem to decompose into rational tangles. We do not know of a way to determine that a non-alternating knot is not algebraic, however.

§1

The motivating example

The reader is referred to [R] for background in knot theory and Dehn surgery. Many of the basic concepts of essential laminations and branched surfaces may be found in [GO].

Typeset by AMS-TEX

Key words and phrases. essential lamination, Seifert surface, Property P.

1 Research supported in part by NSF grant # DMS−9400651 2 Research supported in part by an NSF Postdoctoral Fellowship

The genesis of the constructions we give here is an example due to Ulrich Oertel [Oe], which was analyzed in [Br1]. There a branched surface B was constructed in the complement of a certain tangle T_{0}, in the 3-ball B^{3}; B carried a lamination L which was essential in the complement of any knot obtained by tangle sum with T_{0} (see Figure 1) Further, the lamination remained essential after non-trivial Dehn surgery along any of the knots so constructed.

Figure 1

This lamination and branched surface are, as we shall see, the simplest of a long list of laminations and branched surfaces, that can be associated to any incompressible Seifert surface for a knot. It comes, in fact, from applying our construction to a 2-disk spanning the unknot. We start with this example, in order to motivate the general construction.

One view of Oertel’s construction of the branched surface B is to start with a 2-disk D spanning the unknot, glue a tube to it, and then glue the boundary of the resulting punctured torus to itself, creating a branch curve which runs over the tube (see Figure 2). The reader is invited to verify that the resulting branched surface is isotopic to the one in Figure 1; this is not really central to what follows. If we then string a pair of arcs through both the tube we added to D and the tube created when we glued the branched surface together (Figure 3), and then complete the pair of arcs to a knot K in any way that avoids the branched surface and the two tubes (i.e., the two meridian disks for the tubes), then it is not hard to see [Br1] that the branched surface B is essential in the complement of the knot. Furthermore, since the boundaries of two meridian disks are not isotopic in ∂_{h}N(B) - they live on different components - and each meridian disk intersects our knot exactly once, it follows, primarily from Menasco’s criterion [Me], that our branched surface remains essential after any non-trivial Dehn surgery along the knot K [Br1].

Figure 2 Figure 3

The fact that we will exploit to find our more general construction is that the construction of B did not really use the topology of the Seifert surface D; it was constructed from D by altering it only in a neighborhood of its boundary. We can therefore apply this construction to any surface F with boundary in the 3-sphere, i.e., to any Seifert surface F for a knot K=∂F in the 3-sphere. What we shall see is that if this process is applied to an incompressible Seifert surface for a knot, the lamination we create is persistent for any knot that we construct as above.

§2

The constructions

The construction is completely similar to the one pictured in Figure 2; we simply forget that we can see the entire disk D and focus on a neighborhood of its boundary. Given a Seifert surface F of a knot K¡ a neighborhood of its boundary is an annulus; if we attach a tube to this annulus, running parallel to half of the knot K, and then glue the boundary of F (i.e., K) to a curve running over the tube, and otherwise following the other half of the knot K, we get a branched surface B_{K}; see Figure 4. It is transversely orientable, and so its horizontal boundary splits into a positive part, which we will call ∂_{+}, and a negative part ∂_{−}. This branched surface has a single branch curve, which has no self-intersections, and so it is easy to build a lamination L carried by B_{K} with full support (see [Br1]).

Figure 4

To understand what will happen next, we need a good picture of what the complement of a neighborhood of B_{K}, M_{B} = S^{3}\intN(B_{K}), looks like. Let M_{K} = S^{3}\intN(K). The idea is to think of the modifications described above as taking place in the solid torus neighborhood N(K) of K in S^{3}, so that M_{B}∩M_{K} = M_{K}\intN(F) = M_{F}. M_{F} can be thought of as a sutured manifold, with an annular suture A separating two copies of F, which we will call F_{+} and F_{−}. The Seifert surface F is incompressible precisely when F_{+} and F_{−} are incompressible in M_{F}.

To understand M_{B}, it therefore suffices to understand what N(K)\intN(B) = M_{B}∩N(K) = M_{0} looks like. This is a sutured manifold, with two sutures, one (A_{1}) coming from the branch curve b of B_{K}, and the other (A_{2}) being the annulus ∂N(K)\intN(B) (see Figure 5). By inspection, the complement of the two sutures in ∂M_{0} is a pair of twice-punctured tori (Figure 6); they are each built from (B_{K}∩N(K))\b, which is a 4-punctured sphere, by gluing a pair of boundary circles together (b to c for one, c to d for the other). We will call these two components B_{+} and B_{−}. Each of these punctured tori is compressible in M_{0}, via the meridian disks D_{1},D_{2} in our two tubes (Figure 5). Compressing ∂M_{0} along both of these disks yields a new sutured manifoldM_{1}, with the same set of sutures, whose new positive and negative boundaries are annuli. ∂M_{1} is therefore a torus, and so, since M_{1} is contained in S^{3} and has connected boundary, M_{1} is irreducible. But ∂M_{1} is still compressible; there is a disk D whose boundary meets each suture in a single essential arc (Figure 5). Therefore, M_{1} is a solid torus. In particular, because D crosses exactly two sutures, M_{1} is a \underline{product} sutured manifold (annulus)×I. Therefore, M_{0} is a product sutured manifold with two 1-handles attached, one to the positive boundary and one to the negative boundary. Consequently,

M_{B} ≅ M_{F} ∪M_{0} ≅ M_{F} ∪((annulus)×I∪(2 one-handles)) ≅ M_{F} ∪(2 one-handles)

Therefore, M_{B} = M_{F} with two (3-dimensional) one-handles attached, one having both ends on F_{+}, the other having both ends on F_{−}. ∂_{h}N(B) then consists of the surfaces F_{+} and F_{−}, each with a tube attached (increasing their genera by one). It has two obvious compressing disks, the meridian disks of the two 1-handles, so to kill these we will remove a knot K′ from M_{B} which intersects each of these disks exactly once (see Figure 7). In other words, we think of B_{K} as a branched surface in S^{3}\intN(K′)=M_{K′}.

Figure 5  Figure 6  Figure 7

But it then follows quickly that if F is an \underline{incompressible} Seifert surface for K, then B_{K} is an essential branched surface in M_{K′}, and remains essential after every non-trivial Dehn filling along K′. In other words, B_{K} is persistent for K′.

Most of the arguments are identical the proofs from [Br1]. To show that B_{K} is essential in M_{K′}, we need to know six things:

(1) B_{K} carries a lamination with full support.

This follows, as in [Br1], because B_{K} has a single branch curve whic dies not intersect itself.

(2) B_{K} does not carry a 2-sphere, and B has no disks of contact.

This is because, as in [Br1], B_{K} has only one sector; B_{K}\(its branch curve) is connected. In particular, B_{K} carries no closed surfaces.

(3) B does not carry a compressible torus.

This follows for the same reason as (2).

(4) M_{0} = M_{K′}\int(N(B_{K})) does not have any monogons.

This follows from the fact that B_{K} is transversely orientable.

(5) M_{0} is irreducible.

This is because ∂S^{3}\int(N(B_{K})) = ∂N(B_{K}) is connected, so S^{3}\int(N(B_{K})) is irreducible. Therefore, any reducing 2-sphere in M_{0}⊆S^{3}\int(N(B_{K})) bounds a 3-ball B^{3} in S^{3}\int(N(B_{K})), which therefore contains K′. But then K′ would be null-homotopic in M_{0}, which is impossible since it has non-zero intersection number with each of the two compressing disks D_{1}, D_{2}.

(6) The horizontal boundary ∂_{h}N(B_{K}) of B_{K} is incompressible in M_{0}.

Suppose D is a compressing disk for ∂_{h}N(B_{K}) in M_{0}; in particular, D∩K′ = ∅. Consider D∩D_{1}. This intersection consists of a finite number of circles and arcs. The circles come in two types; those which cut off a disk in D_{1} which contains the intersection of K′ with D_{1}, and those which don’t (see Figure 8). We can remove the circles whose disks don’t meet K′ by a standard disk-swapping argument. The arcs of intersection can be removed by choosing an outermost arc, and using the disk it cuts off from D_{1} to surger D into two disks (see Figure 9). At least one of these is still a compressing disk for ∂_{h}N(B_{K}). That disk has fewer arcs of intersection with D_{1}.

This leaves the circles of interesction which surround the point K′∩D_{1}. But the innermost such circle cuts off a disk in D, which, together with the disk in D_{1} it cuts off, produces a 2-sphere which intersects K′ exactly once. This is impossible, however, in S^{3}.

Therefore, we may assume that D misses D_{1}, and therefore, by symmetry, that it misses D_{2}, as well. This means, then that we can push D out of the two one-handles that were glued onto M_{F} to create M_{B}. In particular, we may think of ∂D as living in F_{+} or F_{−} (wolog F_{+}). But since F_{+} is incompressible in M_{F}, ∂D bounds a disk D′ in F_{+}. If D′ misses the ends of the one-handle that we glued to F_{+}, then D′ is contained in ∂_{h}N(B_{K}), a contradiction. One the other hand, if D′ does contain one or both of these disks, then D separates one or both of these disks from both of the disks where the other one-handle was glued to F_{−} (sse Figure 10). But K′ contains a pair of arcs running from the disks on F_{+} to the disks on F_{−}; if they ran from F_{+} to F_{+}, amd from F_{−} to F_{−}, then K′ would be a link instead of a knot. So at least one of these arcs must intersect D. But this is also a contradiction.

figure 8  Figure 9  Figure 10

Therefore, B_{K} is an essential branched surface in M_{K′}.

§3

Persistence

The two disks D_{1} and D_{2} also allow us to show that B_{K} remains essential after every non-trivial Dehn surgery along K′. The only properties which are not immediate are (5) and (6), because (for (4)) B_{K} remains transversely orientable, and all for of the other properties, N(B_{K}) has not changed, so our previous proof go through without change. Because ∂D_{1} and ∂D_{2} are not isotopic in ∂_{h}N(B_{K}) (they live on distinct components), an argument due to Menasco [Me] shows that ∂_{h}N(B_{K}) remains incompressible afrer non-trivial Dehn surgery along K′. This leaves showing that M_{K′}(p/q)\int(N(B_{K})) = M_{(p,q)} is irreducible for all p/q ≠ 1/0. Again, this argument is essentially the same as that given in [Br1].

Suppose S is a reducing 2-sphere for M_{(p,q)}. We may assume that S intersects (transversely) the core γ of the solid torus glued onto M_{K′}, in the fewest number of points. It is then standard that S\int(N(γ)) = S′ is an incompressible and ∂-incompressible surface in M_{(p,q)}\intN(γ) = M_{K′}\intN(B_{K}) = M_{2}. The curves S′∩∂N(γ) are parallel curves of slope p/q on ∂N(K′)⊆∂M_{2}.

The disks D_{1}, D_{2} meet M_{2} in annuli A_{1}, A_{2}. Consider S′∩A_{i}; it consists of circles and arcs. Trivial circles of itersection can be removed by isotopy. The arcs of intersection cannot meet the boundary component of A_{i} coming from ∂_{h}N(B_{K}), since S misses ∂M_{(p,q)}. These arcs of intersection are therefore boundary parallel in A_{i}, and so can also be removed by isotopy, since S′ is ∂-incompressible. After these isotopies, S′ misses the boundary of A_{i}. If ∂S′ ≠ ∅, then ∂S′⊆∂N(K′) misses a loop in ∂N(γ) which represents a meridion of K′, namely A_{i}∩∂N(γ), and hence consists of meridional loops. So p/q = 1/0, a contradiction.

Therefore ∂S′=∅, in particular, S′ = S⊆M_{2}⊆M_{B}. But since M_{B} is irreducible, S bounds a 3-ball in M_{B}. This 3-ball must intersect, hence contain, K′, since otherwise it is a 3-ball in M_{(p,q)}. But this implies that K′ is null-homotopic in M_{B}, which is impossible since it intersects the disk D_{1} exactly once. So the reducing sphere S cannot exist; M_{(p,q)} is irreducible.

§4

Building the knot K′, without B_{K}

The knots K′ can be readily visualized directly from a picture of the incompressible Seifert surface F of a knot K, without constructing B_{K}. The knots K′ consist of two arcs running through the 1-handles of S^{3}\N(B_{K}), together with two arcs in M_{F} running between these disks on F_{+} and F_{−} where the two one-handles were attached. Referring to Figure 3, the two arcs in the one-handle can be thought of as a pair of subarcs parallel to arc of K, which then pass over and under one another at their ends. The ends then reverse direction and pierce the Seifert surface F in four points (this is required in order to be consistent with the sides of F which the two 1-handles were attached to). These two arcs are then completed to a knot in any way that misses F (and, technically, neighbohoods of the two added crossings). The easiest way to determine which way the two original arcs cross each other is to assign a ‘+’ and ‘−’ side to F, and think of one arc remaining slightly on the ‘−’ side of F, and the other remaining slightly on the ‘+’ side of F. This gives the correct ‘parity’ to the two crossings. The ends of the arcs from K can also be assigned a ‘+’ and ‘−’, marking which side of F they are emerging from; in order to obtain a knot K′, one then simply needs to join the four ends by arcs missing F, so that the +’s are joined to the −’s; see Figure 11.

Figure 11

§5 Some examples

There are (at least) three ways to obtain an ioncompressible Seifert surface for a knot K:

(1) run Seifert’s algorithm on an alternating projection for a knot [Cr];

(2) find a surface whose genus = \frac{1}{2}(span of the Alexander polynomial for the knot)

[S] - in general, genus(K)≥\frac{1}{2}(span);

(3) build a once-punctured torus whose boundary you \underline{know} (for other reasons) is a non-trivial knot.

Note: all of these in fact give \underline{least} \underline{genus} Seifert surfaces.

If you have a surface which you suspect is least genus (hence incompressible), Gabai’s theory of sutured manifold heirarchies [Ga] can, in principle, prove it is. If you have a surface which you suspect is incompressible, Haken’s normal surface theory (see [JO]) can, in principle, prove or disprove it.

We illustrate each of these techniques with a few examples: see the figures below.

Figure 12 − figure ???

§6

Generalizations

§7

Concluding remarks

It has long been conjectured that all non-trivial knots in S^{3} have Property P, that is, that non-trivial surgery on the knot can never yield a simply-connected manifold. On the other hand, not all knots have strong Property P: torus knots, for example, have surgeries with finite fundamental group, and cabled knots have reducible surgeries. Both of these properties preclude the surgery manifold from having universal cover ℝ^{3}.

The constructions we have described here represent only one very basic sort of branched surface in S^{3}, yet they genereate a wealth of examples of knots admitting persistent laminations, and hence having strong Property P. It is clear that much more can be gained, and learned, by applying this construction to other methods of building branched surfaces.

One question that deserves an answer is whether or not \underline{every} knot with strong Property P admits a persistent lamination. This is probably false, most likely because it happens to be the case [Br2],[Cl] that most exceptional Seifert-fibered spaces do not admit essential laminations, although they do have universal cover ℝ^{3}. It is in fact not known that every non-torus alternating knot admits a persistent lamination, although as mentioned in the introduction, they do have strong Property P. There is at least one way to show that a knot does not admit a persistent lamination [BNR], but it requiresknowing that one of the surgery manifolds is the ‘right’ kind of graph manifold. Whether or not this is the case for any alternating knot might be an interesting topic of study.

Another question which we \underline{can} answer is whether or not, for all of the knots we can construct by this method, the surgery manifolds are in fact all Haken, i.e., contain an incompressible closed surface. It turns out that this construction can generate a 2-bridge knot from certain other 2-bridge knots (see Figure *****); since a 2-bridge knot exterior contains no closed, non-∂-parallel incompressible surfaces [HT], it follows [Ha] that all but finitely many Dehn surgeries on these knots are non-Haken.

Figure ???+1

References

[Br1] M. Brittenham, Persistently laminar tangles, preprint.

[Br2] M. Brittenham, Essential laminations in Seifert-fibered spaces, Topology 32 (1993), 61-85.

[BNR] M. Brittenham, R. Naimi, and R. Roberts, Graph manifolds and taut foliations, preprint.

[Cl] E. Claus, Essential laminations in closed Seifert-fibered spaces, Thesis, Univ. of Texas at Austin (1991).

[Cr] R.H. Crowell, Genus of alternating link types, Annals of Mathematics 69 (1959), 258-275.

[De] C. Delman, Constructing essential laminations which survive all Dehn surgeries, Topology and its Appl. ???.

[DR] C. Delman and R. Roberts, Constructing essential laminations in alternatig knot exteriors, in preparation.

[Ga] D. Gabai, Genera of the arborescent links, Memoirs of the AMS 339 (1986), 1-98.

[GO] D. Gabai and U. Oertel, Essential laminations in 3-manifolds, Annals of Mathematics 130 (1989), 41-73.

[Ha] A. Hatcher, On the boundary curves of incompressible surfaces, Pac J Math 99 (1982), 373-377.

[HT] A. Hatcher and W. Thurston, Incompressible surfaces in 2-bridge knot complements, Inv Math 79 (1985), 225-246.

[JO] W. Jaco and U. Oertel, An algorithm to decide if a 3-manifold is a Haken manifold, Topology 23 (1984), 195-209.

[Me] W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984), 37-44.

[Na] R. Naimi, Essential lainations in 3-manifolds obtained by surgery on 2-bridge knots, Contemp. Math. 164 (1994), 183-186.

[Oe] U. Oertel, personal communication.

[R] D. Rolfsen, Knots and Links, Publish or Perish Press, 1976.

[S] H. Seifert, Über das Geschlecht von Knoten, Math. Annalen 110 (1934), 571-592.

[Wu] Y.-Q. Wu, Dehn surgery on algebraic knots, preprint.

Department of Mathematics, Box 30001 Dept. 3MB, New Mexico State University, Las Cruces, NM 88003

E-mail address: britten@math.nmsu.edu

Department of Mathematics, Washington University, St. Louis, MO 63130 E-mail address: roberts@math.wustl.edu