detecting incompressibility of boundary in 3-manifolds
TRANSCRIPT
Detecting Incompressibility of Boundary
in 3-Manifolds
COLIN ADAMSDepartment of Mathematics, Williams College, Williamstown, MA 01267, U.S.A.e-mail: [email protected]
(Received: 21 March 2001; in final form: 15 March 2002)
Abstract. A construction is presented which can be utilized to prove incompressibility of
boundary in a 3-manifold W. One constructs a new 3-manifold DW by doubling W along asubsurface in its boundary. If DW is hyperbolic, and if W has compressible boundary, thenDW must have a longitude of ‘length’ less than 4. This can be applied to show that an arc
a that is a candidate for an unknotting tunnel in a 3-manifold cannot be an unknotting tunnel.It can also be used to show that a ‘tubed surface’ is incompressible. For knot and link com-plements in S3, and a an unknotting tunnel, DW is almost always hyperbolic. Empirically, this
construction appears to provide a surprisingly effective procedure for demonstrating that spe-cific arcs are not unknotting tunnels.
Mathematics Subject Classification (2000). 57M50.
Key words. hyperbolic 3-manifold, unknotting tunnel, boundary incompressibility.
1. Introduction
Given a specific compact orientable 3-manifold W with boundary, it is important to
know whether the boundary is incompressible. If not, one could compress it to sim-
plify the manifold.
In theory, one can utilize a triangulation of the surface and normal surface theory
to determine the incompressibility of the boundary. However, this can be a difficult
and computationally intensive procedure. Here, a method is presented which will
often demonstrate easily the incompressibility of the boundary of the manifold.
One creates a new manifold DW that is obtained by doubling the original manifold
across a particular subsurface in the boundary. If DW is hyperbolic, it will be of
finite volume with a finite set of cusps. If the boundary of W is compressible, then
on the boundary of any set of cusps in DW with disjoint interiors, there must be
a nontrivial closed curve of length less than 4. No such curve immediately implies
that the boundary of W is incompressible. The SNAPPEA computer program for
investigating hyperbolic structures on 3-manifolds, written by Jeffrey Weeks (cf.
[18]) can be used to determine whether or not there are nontrivial closed curves of
length less than 4 in the boundary of a disjoint set of cusps in DW. One inputs
the manifold DW into the program, which then computes the hyperbolic structure
Geometriae Dedicata 99: 47–60, 2003. 47# 2003 Kluwer Academic Publishers. Printed in the Netherlands.
and displays the cusps. From this, one easily determines whether or not there are
curves of length less than 4 in the boundaries.
We will be particularly interested in the applications of this result to unknotting
tunnels. Let M be a compact 3-manifold with one or two toroidal boundary compo-
nents. A properly embedded arc is said to be an unknotting tunnel if the complement
of a regular neighborhood of the arc is a genus two handlebody. The handlebody,
which has comprssible boundary, will correspond to our 3-mainfold W.
A manifold with an unknotting tunnel is said to be of tunnel number one. More
generally, a compact manifold with boundary is said to be tunnel number n if there
exists a set of n disjoint properly embedded arcs such that the complement of an
open regular neighborhood of the set of arcs is a handlebody, and no such set with
fewer arcs exists. Such a set of arcs is called an unknotting tunnel system. A knot or
link in the 3-sphere is said to be tunnel number n if its exterior is tunnel number n.
A good reference for unknotting tunnels is [16].
Torus knots are tunnel number one and their unknotting tunnels were classified in
[5]. Thinking of a torus knot exterior as obtained by gluing two solid tori together
along an annulus in each of their boundaries, the unknotting tunnels correspond
to an arc in the annulus cutting the annulus into a disk and an arc that starts at
the knot, drills in to the core curve of one of the solid tori, travels around the core
curve and then goes back out to the knot.
Two bridge knots and links are tunnel number one. The unknotting tunnels for
two-bridge links (as opposed to knots) were classified in [4] and [10] and consist only
of the so-called upper and lower tunnels, horizontal arcs that connect the two max-
ima and the two minima when the 2-bridge knot or link is drawn so that the two
maxima occur at the same height at the top and the two minima occur at the same
height at the bottom. The unknotting tunnels for two-bridge knots are classified in
[9], and consist of the upper and lower tunnels and their so-called duals.
Tunnel number one knots in S3 are known to be prime. (cf. [14, 17]). The tunnel
number one knots and links in the 3-sphere with nonsimple complements have been
classified along with their unknotting tunnels in [6, 7, 12]. In [13], the authors utilize
2-fold covers to determine the tunnel numbers of prime knots through ten crossings,
showing that all but 87 of the 249 knots are tunnel number one, with the remainder
of tunnel number two. In the case of 2-cusped hyperbolic 3-manifolds, it was shown
in [2] that unknotting tunnels are isotopic to geodesics of length (relative to a max-
imal set of cusps) less than ln(4). Hence, for such a manifold, one need only check a
finite number of candidates to obtain a complete classification of the unknotting
tunnels.
However, determining whether or not a candidate arc in a 3-manifold is an
unknotting tunnel is generally difficult. In this paper, we will describe a technique
and its variations that seem very effective in showing that a particular candidate is
not an unknotting tunnel. Although it utilizes hyperbolic 3-manifold theory, there
is no assumption that the original manifold is hyperbolic. The constructions are
described below.
48 COLIN ADAMS
A compact manifold with torus boundary components will be said to be hyper-
bolic if there is a complete hyperbolic metric on its interior. Such a manifold will
have a finite hyperbolic volume. Because it is hyperbolic, there is a covering map
p: H3 ! intðMÞ. W.Thurston showed that a compact orientable 3-manifold is hyper-
bolic if and only if it is irreducible, boundary irreducible and it does not contain any
essential tori or annuli.
Each torus boundary component corresponds to a cusp in the hyperbolic struc-
ture. Expanding that cusp to be as large as possible while preserving the fact it
has embedded interior yields a so-called maximal cusp (see [1]). A set of cusps that
do not overlap in their interiors and such that any expansion of one of the cusps
would cause such an overlap is called a maximal set of cusps. Note that if the bound-
ary of any maximal set of cusps does not contain a nontrivial closed curve of length
less than 4, then no set of cusps with disjoint interiors contains a nontrivial closed
curve of length less than 4 in its boundary.
Given a 3-manifold W with boundary and a surface F contained in the boundary,
the double ofW along F is the 3-manifold DW that results if we reflectW through F
to obtain a copy RW and take the union of W and RW, which share F.
THEOREM 1.1. Let W be a 3-manifold with compressible boundary. Let fc1; . . . ; ckg
be a system of disjoint nontrivial simple closed curves on @W. Let DW be the mani-
fold that results if W is doubled along the surface F in its boundary given
by F ¼ @W � [NðciÞ. Suppose that DW is hyperbolic. Then it has the following
properties:
ðiÞ The surface F is the totally geodesic fixed-point set of an orientation reversing invo-
lution R.
ðiiÞ There exists a longitude curve of length less than 4 in the boundary of some cusp in
any set of cusps of DW with disjoint interiors.
The first conclusion is immediate. It is the second conclusion in which we are inter-
ested. It will have applications to unknotting tunnels and to incompressibility of
tubed surfaces. Although the hypothesis that the resulting manifold DW be hyper-
bolic seems restrictive, we will see that often, it is not.
If M is a compact orientable 3-manifold with boundary consisting of a nonempty
set of tori, and if fa1; . . . ; ang is an unknotting tunnel system for M, then
W ¼ M � [NðaiÞ is a genus g handlebody. Hence it has compressible boundary. If
we apply the construction above, and DW is hyperbolic, there will be a nontrivial
closed curve of length less than 4 in the boundary of any set of cusps with disjoint
interiors. However, we have lots of choice in the curves fc1; . . . ; ckg.
We now give three explicit choices that result from the general construction. The
first two are relevant to unknotting tunnels and the third to tubing surfaces.
We describe the first construction for one unknotting tunnel a in a manifold M,
although it can also be applied to an unknotting tunnel system. Note that in our
situation, @M must consist of one or two tori. Choose two disjoint simple nontrivial
DETECTING INCOMPRESSIBILITY OF BOUNDARY IN 3-MANIFOLDS 49
closed curves b1 and b2 on the boundary of the manifold such that each passesthrough one of the endpoints of a. They will be parallel if both endpoints of a areon the same boundary component. Remove an open regular neighborhood of
a [ b1 [ b2 fromM, calling the resulting 3-manifoldW. Since a is an unknotting tun-nel, W must be a handlebody. That part of the boundary of W that was not part of
the boundary of M is a 4-holed sphere, call it F. Double the manifold W along F to
obtain a new manifold, denoted D1W. This corresponds to a choice of curves c1 and
c2 in the boundary of W that are core curves of the two annuli in @W � F.
The manifold D1W will have two torus boundary components, and the obvious
reflective involution. In the case that D1W is hyperbolic, the four-holed sphere F will
be the fixed point set of the reflective involution and will thereforre appear in the
hyperbolic structure as a totally geodesic surface.
For convenience, call the intersection of the four-holed sphere with the bound-
ary of either cusp a meridian and a perpendicular curve on the cusp boundary a
longitude. Note that the reflective involution implies that there does exist a per-
pendicular curve in the maximal cusp boundary. In the case that we start with a
knot or link exterior in the 3-sphere, and we choose b1 and b2 to be meridians inthe traditional sense, D1W will be the exterior of a 2-component link in the
3-sphere with our meridian and longitude corresponding to the traditional meri-
dian and longitude.
Since each of the two cusps in the manifold puncture the four-punctured sphere
twice, upper bounds on the size of cusps in hyperbolic surfaces from [3] imply that
the meridian of each cusp, as measured in the boundary of the corresponding maxi-
mal cusp must have length at most 5. We will see an example where 5 is realized.
From this upper bound on the length of the meridian, we obtain the following
corollary.
COROLLARY 1.2. Let a be an unknotting tunnel in a 3-manifold M such that D1W is
hyperbolic. Then at least one of the cusps in any set of cusps with disjoint interiors in
D1W must have volume strictly less than 10.
In the case that the arc a is an unknotting tunnel in a knot or link exterior in S3, it
turns out that D1W is almost always a hyperbolic manifold.
THEOREM 1.3. Let a be an unknotting tunnel for a nontrivial knot or link in the
3-sphere and let b1 and b2 be meridian curves. Then D1W is hyperbolic if and only if a is
not the upper or lower tunnel for a 2-bridge knot or link. When D1W is hyperbolic, the
longitude of some cusp boundary in any set of cusps with disjoint interiors has length
strictly less than 4.
By Theorem 1.1, D1W must always have a reflective involution R. In the case
when there is only one unknotting tunnel, rather than an unknotting tunnel system,
D1W also has an orientation preserving involution J as follows. A genus two handle-
50 COLIN ADAMS
body has an involution that extends to the regular neighborhood of the unknotting
tunnel with 1-dimensional fixed point set intersecting the boundary of the regular
neighborhood of the unknotting tunnel at two points. Upon doubling, we obtain
an involution with circular fixed point set B intersecting F at two points.
In Figure 1, we construct D1W for the knot 52, using one of its dual tunnels. In this
case, the longitude of the maximal cusp corresponding to the component bounding
the twice-punctured disk has length 2. As a side note, this component has meridian
of length exactly 5, the upper bound. It is worth noting that the resulting manifold
D1W also corresponds to known unknotting tunnels for 810 and 820. In fact, any
given D1W is always generated by an infinite set of manifold and candidate unknot-
ting tunnel pairs.
The upper and lower tunnels of the two-bridge knots and links are treated as
exceptional cases in the statement of Theorem 1.3, as in that case, D1W becomes
the exterior of the trivial link of two components. The construction automatically
removes any rational tangles around the unknotting tunnel, and in the case of the
upper and lower tunnels for 2-bridge knots and links, that leaves no knotting.
One might wonder whether a longitude of length less than 4 in D1W forces a to bean unknotting tunnel. In the vast majority of examples of arcs that were not unknot-
ting tunnels, SNAPPEA (the hyperbolic structures program written by Jeffrey
Weeks [18]) generated a D1W with all longitudes of length at least 4. However, there
is at least one example with longitude less than 4 which does not correspond to an
unknotting tunnel.
In fact, many of the examples of arcs that are not unknotting tunnels have a long-
itude of length exactly 4 for the following reason. For many of the arcs under
consideration, there is an annulus properly embedded in W which touches F along
an arc in its boundary and, hence, which doubles to a twice-punctured disk in
D1W. This occurs for the example in Figure 1 and for the dual tunnels in any of
the two-bridge knot complements, for instance. Hence, there is an incompressible
boundary-incompressible thrice-punctured sphere properly embedded in D1W, one
puncture of which corresponds to a longitude of one of the cusps. As is well known,
the thrice-punctured sphere is totally geodesic. But a maximal cusp in the thrice-
punctured sphere has boundary of length exactly 4. Hence, when this cusp in
D1W is maximized, either its point of tangency with itself occurs in the thrice-
Figure 1. Construction of D1W for dual tunnel in 52 knot complement.
DETECTING INCOMPRESSIBILITY OF BOUNDARY IN 3-MANIFOLDS 51
punctured sphere, in which case the length of this longitude is exactly 4, or the point
of tangency of this cusp with itself does not occur in the thrice-punctured sphere, in
which case the length of the longitude is strictly less than 4.
The construction for D1W generalizes to an n-arc unknotting tunnel system in a
manifold M. In the case M is a link exterior in the 3-sphere, D1W is the exterior
of a 2n-component link in the connected sum of n � 1 copies of S2 S1. Again, at
least one cusp boundary in any set of cusps with disjoint interiors must have longi-
tude of length strictly less than 4.
We now describe a second construction based on the general framework. Let
fa1; . . . ; ang be an unknotting tunnel system for the manifold M, which has non-
empty toroidal boundary components. Let W be the complement of the open neigh-
borhoods of the unknotting tunnel system. Let F ¼ @W � [@NðaiÞ. Let D2W be the
double of W along F. Note that D2W is obtained by doubling M along its toroidal
boundaries and then removing neighborhoods of the doubled arcs in the system.
When D2W is hyperbolic, at least one maximal cusp must have longitude of length
less than 4. In the case that there is a single unknotting tunnel and M is a knot ex-
terior in the 3-sphere, D2W is usually hyperbolic:
THEOREM 1.4. If a is an unknotting tunnel for a nontrivial knot in the 3-sphere, then
D2W is hyperbolic if and only if a is not an unknotting tunnel for a torus knot or cable
knot. When D2W is hyperbolic, then a longitude on the single maximal cusp must have
length strictly less than 4.
Note that the cable knots that are tunnel number one are a relatively restricted
class, corresponding to certain cables on torus knots (cf. [6, 12]). Although the fact
that D2W has only one cusp is an advantage to this construction, the disadvantage is
that D2W will not be a knot or link exterior even though M may be one. Note that
between the two constructions, the only tunnel number one knots in S3 such that
their doubles under each of these constructions are not hyperbolic are the 2-braid
knots. In all other cases, either D1W or D2W will be hyperbolic.
The third construction was suggested by William Menasco. Let G be a properly
embedded incompressible boundary incompressible surface with boundary in a com-
pact connected orientable manifold W with toroidal boundary components. From
such a surface, one would like to construct a closed incompressible surface by adding
tubes that run around the toroidal boundary components from one boundary com-
ponent of G to another. See [8, 11, 15] for more on this tubing operation. One needs
to know that the resulting surface is incompressible.
Let G0 be the surface obtained by tubing together pairs of boundary components
of G by disjoint annuli A1; . . . ;An in @W. Let W0 be the manifold that is obtained
when W is cut open along G. Note that it may have more than one component.
52 COLIN ADAMS
Let D3W be the manifold obtained when W0 is doubled along the copies of G in its
boundary.
THEOREM 1.5. Let G be a properly embedded orientable incompressible boundary-
incompressible surface in a connected orientable 3-manifold W, Let G0 be obtained by
tubing together pairs of boundary components of G. If D3W is hyperbolic and G0 is
compressible, then there must be a longitude with length less than 4 in the cusp
boundary in any set of cusps with disjoint interiors in one of the components of D3W.
The remainder of the paper is devoted to proofs of the above theorems and a
corollary.
2. Proofs
THEOREM 1.1. Let W be a 3-manifold with compressible boundary. Let fc1; . . . ; ckg
be a system of disjoint nontrivial simple closed curves on @W. Let DW be the manifold
that results if W is doubled along the surface F in its boundary given by
F ¼ @W � [NðciÞÞ. Suppose that DW is hyperbolic. Then it has the following properties:
ðiÞ The surface F is the totally geodesic fixed-point set of an orientation reversing invo-
lution R.
ðiiÞ There exists a longitude curve of length less than 4 in the boundary of some cusp in
any set of cusps of DW with disjoint interiors.
Proof. Part (i) is immediate from the construction, since the fixed point set of an
involution must be totally geodesic. It remains to prove Part (ii).
Assume that we have fixed a choice of cusps fC1;C2; . . . ;Cng with disjoint inter-
iors. Lifting the manifold DW to hyperbolic 3-space, fC1;C2; . . . ;Cng lift to a set
of horoballs with disjoint interiors, and the surface F lifts to a set of totally geodesic
planes, with boundaries passing through the centers of the horoballs. Since W has
compressible boundary, there exists a compressing disk D, which lifts to a disk D0
with interior in the complement of the horoballs and with boundary in the union
of the boundaries of the horoballs and the totally geodesic planes. The boundary
of D cannot be contained entirely in F, or in the boundary of one of the cusps, as
they are each incompressible. Hence @D0 must lie in p�1ð@C1 [ � � � @Cn [ FÞ, and pass
through an alternating sequence of horospheres and totally geodesic planes. Assume
we have isotoped D to minimize the number of times that @D0 intersects horospheres
and totally geodesic planes in p�1ð@C1 [ � � � @Cn [ FÞ. Then there must be a finite
sequence of horospheres alternating with geodesic planes, each of which intersects
the next in the sequence, such that the sequence begins and ends at the same horo-
sphere. Choose that horosphere to be centered at infinity in the upper-half-space
DETECTING INCOMPRESSIBILITY OF BOUNDARY IN 3-MANIFOLDS 53
model, and then take a smallest horosphere H in the cyclic sequence (smallest in the
Euclidean sense). If there is a subsequence of equally small horospheres in the
sequence, choose H on the end of the subsequence. In order that one of its neighbor-
ing horospheres be at least as big as (or bigger than) H, it must be the case that
the distance on H from its peak to its intersection by the connecting totally geodesic
plane be less than or equal to (or less than) 1. Hence the distance on the horosphere
between the two geodesic planes that connect this horosphere to its two neighboring
horospheres must be strictly less than 2. However, that distance appears in the cusp
boundary in the manifold as exactly half a longitude of one of the cusps, beginning
and ending on F. When we reflect through F this path becomes a longitude in the
boundary of one of the cusps of length less than 4. &
Given a properly embedded arc a in a manifold M that is a candidate for an
unknotting tunnel, then in order to show that a is not an unknotting tunnel, it isenough to check that if W ¼ M � NðaÞ, and D1W is hyperbolic, then there exists a
choice of cusps in D1W with disjoint interiors, neither of which has a longitude
shorter than 4. Although not as effective a means of eliminating unknotting tunnel
candidates, one further has:
COROLLARY 1.2. Let a be an unknotting tunnel in a 3-manifold M such that D1W is
hyperbolic. Then at least one of the cusps in any set of cusps with disjoint interiors in
D1W must have volume strictly less than 10.
Proof. The area contained within a maximal set of cusps in a totally geodesic
four-punctured sphere is at most 12 (cf. [3]). Any single cusp in the set has an area of
at least 1. Since either cusp in D1W intersects F twice, and each intersection must
have the same area, 5 is an upper bound on the meridian of either cusp. At least one
cusp must have longitude strictly less than 4. This cusp then has area of its boundary
less than 20 and hence volume less than 10. &
THEOREM 1.3. Let a be an unknotting tunnel for a nontrivial knot or link in the
3-sphere and let b1 and b2 be meridian curves. Then D1W is hyperbolic if and only if a is
not the upper or lower tunnel for a 2-bridge knot or link. When D1W is hyperbolic, the
longitude of some cusp boundary in any set of cusps with disjoint interiors has length
strictly less than 4.
Proof. Let M be the original knot or link exterior. We consider W as a handle-
body contained in D1W, and RW its reflection across F, the 4-holed sphere. Let F 0 be
the sphere obtained in S3 by capping off the four holes of F.
We first note that F is incompressible and boundary-incompressible in D1W. It is
enough to show that as a surface on the boundary ofWðor RW Þ, it is incompressible
and boundary-incompressible. Note that W is a cube-with-wormholes contained in
S3. For convenience, denote the cube in S3, which is bounded by F 0, by C and the
two arcs, the removal of whose neighborhood generates the wormholes, byV1 andV2.
54 COLIN ADAMS
If F compressed inW, then the boundary of the compressing disk D must separate
two of the holes in F from the other two. Hence D separates C into two balls, each of
which contains one of V1 and V2. Then V1 and V2 must be trivally knotted in each
ball since, otherwise, we could construct an essential torus in W, which cannot exist
in a handlebody. However, this forces V1 and V2 to generate a rational tangle in C.
This means that a is the upper or lower tunnel in a 2-bridge knot or link exterior andDW is the exterior of the trivial two- component link, a case we have excluded from
consideration.
If F boundary-compressed, it must do so in W. This would imply that V1 and V2can be isotoped into F. That in turn implies F is compressible, a contradiction.
In order to show that D1W is hyperbolic, work of Thurston shows it suffices to
demonstrate that D1W is irreducible, boundary-irreducible, and it contains no essen-
tial tori or annuli.
Suppose D1W is reducible. Let S be a reducing sphere. Then S must intersect
F, since the handlebody W is irreducible. After minimizing the intersections of S
and F, we can take an innermost intersection curve on S. This yields a disk D0 in
W with boundary of F. By incompressibility of F, the boundary of the disk
must be trivial in F, and by irreducibility of W, the disk D0 the disk D0 could
have been isotoped to eliminate another intersection curve, a contradiction to
minimality.
Suppose D1W is boundary-reducible. Then there is a disk D embedded in DW with
nontrivial boundary on one of the two toroidal boundary components. D must inter-
sect F as otherwise, D exists in M, a contradiction to the incompressibility of a mer-
idian in a knot or link exterior in the 3-sphere. By incompressibility of F, the closed
intersection curves in D \ F can be eliminated. Choose D to have the minimum pos-
sible number of intersection curves with F. Taking an outermost intersection arc in
D, we obtain a sub-disk of D that lies entirely inW and that has boundary consisting
of one arc in @W � F and one arc in F. This contradicts the boundary-incompressi-
bility of F.
Suppose there exists an essential torus T in D1W. If T does not intersect F, then it
lies entirely in W, and is essential there. But this is not possible for a handlebody.
On the other hand, if T intersects F, then we can eliminate trivial intersection
curves on T and F. Assume that we have minimized the number of remaining inter-
section curves. Then T intersects W in a set of annuli.
Let A be such an annulus. Suppose first that A is knotted in S3, in the sense that
the torus formed from A and an annulus in F 0 with the same boundary is knotted
in S3. Then A splits C into a knot exterior E and a ball B. Since A is incompres-
sible, B must contain at least one of V1 and V2. Since A cannot split a knot exter-
ior from W, E must contain one of V1 and V2 as well, say V2. Since W is a
handlebody, it cannot contain a knot exterior with incompressible boundary. That
is to say, V1 must be prime, i.e. unknotted in B. Hence, A is parallel to @NðV1Þ
through B.
DETECTING INCOMPRESSIBILITY OF BOUNDARY IN 3-MANIFOLDS 55
On the other hand, if A is unknotted, then A splits the ball bounded by F 0 into a
ball B0 and a solid torus S0. Then B0 must contain one of V1 and V2 by incompres-
sibility of A, say V1, and S0 must contain V2 as otherwise, the number of intersection
curves of T with F could be lowered. By the fact W is a handlebody, V1 must be
unknotted in the ball. Hence, A is parallel into @NðV1Þ.
Thus T intersects W in a set of annuli, each of which is individually parallel to
@NðV1Þ or @NðV2Þ. Then, in D1W;T is made up of annuli to each side of F that
are parallel to @NðV1Þ, @NðV2Þ and their reflections through F. If a given annulus
A in W is parallel to @NðV1Þ then the annulus that is its continuation in RW at
one of its boundary components must be parallel to R@NðV1Þ, since this bound-
ary component wraps once around the hole corresponding to R@NðV1Þ. Thus T is
boundary-parallel in D1W, contradicting our assumption that it is essential.
Suppose that D1W contains an essential annulus A. We first show it must intersect
F. Otherwise it would be contained entirely inW or RW, sayW. Then A has bound-
aries that are meridians on R@NðV1Þ or R@NðV2Þ. But then it becomes a sphere when
capped off with meridianal disks from NðV1Þ and/or NðV2Þ, meaning that both of its
boundary components are meridianal curves on the same strand, say V1, and that V1cannot be knotted inside the ball cut off by A. But then A is parallel to @NðV1Þ, con-
tradicting the fact A is not boundary-parallel.
So A must intersect F. After removing all trivial intersection curves in A \ F, all
the remaining curves are either closed or arcs. If they are all closed, we can choose
an annulus A0 on A containing no other intersection curves such that one of its
boundary components, call it g, is a boundary of A and the other is an intersection
curve with F. Then g is a meridian on @NðV1Þ or @NðV2Þ, say @NðV1Þ for convenience.
Then A0 capped off with a meridian disk in NðV1Þ is a disk and therefore it separates
the ball bounded by F into two balls. Again, V1 must be unknotted to the side of A0
that does not contain V2 since a handlebody does not contain a knot exterior with
incompressible boundary. We can isotope A through that side to eliminate an inter-
section curve. Repeating this process, we eventually eliminate all intersection curves,
a contradiction.
Otherwise, A \ F consists of essential arcs in A. Hence, there exists a disk D con-
tained in A, such that its boundary consists of four arcs, two contained in F and two
contained in @W � F. The double of D is an annulus in D1W. Suppose first that for
the two arcs that are contained in @W � F, one is contained in @NðV1Þ and one con-
tained in @NðV2Þ. Then each arc must begin at one boundary component of @NðViÞ
and end at the other. Hence V1 and V2 are parallel in C. Since they cannot be knot-
ted, as that would create an essential torus in W, it must be the case that V1 and V2are two unknotted parallel arcs in C. HenceM is a 2-bridge knot or link and a is theupper or lower tunnel.
Suppose that both of the arcs on the boundary of D that are in @W � F are in
@NðV1Þ, for instance. Then D cuts a ball from W. Since A is boundary-incompres-
sible, V2 must be contained in the ball. Since W contains no essential tori,
V2 must be trivially knotted within the ball. This again implies V1 and V2 are
56 COLIN ADAMS
parallel, and trivial, and that M is a 2-bridge knot or link exterior and a is theupper or lower tunnel. &
THEOREM 1.4. If a is an unknotting tunnel for a nontrivial knot in the 3-sphere, then
D2W is hyperbolic if and only if a is not an unknotting tunnel for a torus knot or cable
knot. When D2W is hyperbolic, then a longitude on the single maximal cusp must have
length strictly less than 4.
Proof. Let M be the original knot exterior. First note that if a is an unknottingtunnel for a torus knot or a cable knot, then by the classification of such unknotting
tunnels in [5, 6, 12], one can see that there is either a disk or annulus in W that
doubles to an essential annulus or essential torus in D2W, and it is therefore not
hyperbolic.
We now prove the theorem in the other direction. We first show that F is
incompressible and boundary-incompressible in D2W. It is incompressible since
@M is incompressible in M. If F boundary-compressed, then there would be a
disk D with one arc of its boundary in F and the remaining arc in @NðaÞ. Thena would be isotopic into F. However, this implies M itself is a solid torus, a
contradiction.
Now, we will show that D2W is irreducible, boundary-irreducible, and contains
no essential tori or annuli. That it is irreducible follows immediately from the fact
M is irreducible and F is incompressible. If it boundary-reduced, and the reducing
disk D did not intersect F, there would be a nonseparating sphere in M, which
cannot occur in S3. Assume D has minimal intersections with F. Take as intersec-
tion curve in F \ D that is outermost on D. This generates a disk D0 with one
boundary in @NðaÞ and the other in F, contradicting the boundary incompressi-
bility of F.
Suppose T is an essential torus in D2W. It must intersect F since there are no essen-
tial tori in a handlebody. After minimizing intersections, we can assume all intersec-
tion curves are nontrivial on F and T, and F cuts T into a set of annuli. Let A be such
an annulus in W. It is incompressible in W, but since a handlebody cannot contain
an essential annulus, it must be boundary-compressible in W. Note that both of
its boundary curves must either be trivial or nontrivial on @M by incompressibility
of @M.
Suppose first that the boundaries of A are trivial on @M. Capping each boundary
component off with a disk in @M, we obtain a sphere which must bound a ball inM.
If a is contained in the ball, it is unknotted in the ball since a handlebody cannot con-tain a knot exterior with incompressible boundary. So A is parallel through the ball
to @N(a).If a is not in the ball, then since A is incompressible, the two boundary compo-
nents of A are concentric on @M, and at least one of the ends of a is containedinside the disk on @M bounded by the inner boundary component of A. Then A is
boundary-parallel in W, a contradiction to the minimality of intersection curves
between T and F.
DETECTING INCOMPRESSIBILITY OF BOUNDARY IN 3-MANIFOLDS 57
Suppose now that the boundary curves of A are nontrivial on @M. If they are both
meridians, then by capping off A with meridian disks, we have a sphere bounding
two balls in S3. If to one side, the arc of K contained in the ball is unknotted, then
A is parallel to @M in M. If it remains parallel in W, we have a contradiction to
minimality of intersections of F and T. If it does not, then the ball also contains
a, a contradiction to a being a valid unknotting tunnel. If the arcs of K are knottedto each side, the knot is composite, a contradiction to being tunnel number one (cf.
[14, 19]).
If the boundary curves of A are not meridians, then we will show that they consist
of exactly one longitude. First note that A is either boundary-compressible or bound-
ary-incompressible in M. In the first case, A is boundary-parallel in M. Then by
minimality of intersections of F and T, a would have to be contained in the solidtorus that gives the isotopy of A to @M. But this contradicts the fact a is an unknot-ting tunnel. So A is boundary-incompressible in M.
Suppose that the curves in @A are not longitudinal on @N(K). Let b be the corecurve of A. Then b is a (p, q)-cable on K, q5 2. Each half of the annulus A realizes
the cabling annulus for this (p, q)-cable on K. But such an annulus is unique up to
isotopy, so the one half of the annulus is isotopic to the other half. This yields a
boundary-compression of A, a contradiction. But if @A is longitudinal on @N(K),
then K is a torus knot or cable knot.
So the only possibility other than K being a torus knot or cable knot is that each of
the annuli that make up T are parallel into @N(a) or its reflection. Hence, there mustbe exactly two such annuli, and T is boundary-parallel in D2W, a contradiction.
Suppose now that A is an essential annulus in D2W. First suppose that A does not
intersect F. Then both boundary components are meridians on @N(a). Capping offthe two boundaries of A by meridian disks in N(a) yields a sphere. Since a handle-body cannot contain a knot exterior with incompressible boundary, a is unknottedin the sphere, and A must be parallel into @N(a) in W, a contradiction.
Now suppose that A intersects F in a minimal set of simple closed curves. Since
each boundary component of A must be a meridian of N(a), which is trivial in M,
A must intersect F in a trivial curve as well. This intersection curve must contain
an endpoint of a. Again, capping A off with disks yields a sphere inM, which bounds
a ball in M. That part of a that is inside the sphere must be unknotted. Then we canslide one component of @A down along @N(a) to lower the number of intersectioncurves of A\F, a contradiction to minimality.
Finally, suppose that the intersection curves of A with F are essential arcs in A.
They cut A into a set of disks, each of which has two arcs in its boundary in F
and the remaining two arcs in its boundary in @N(a). Let D be such a disk in W.
There are three possibilities. First, @D could be a trivial loop on @W. Then A is
boundary-compressible.
Second, D could be a separating compressing disk in the handlebody W. Then D
cuts W into two solid tori. There is an annulus contained in N(a)[D that cuts M
into two solid tori. The only nontrivial knot exteriors in the 3-sphere that consist
58 COLIN ADAMS
of two solid tori glued together along an annulus in the boundary of each are the
torus knot exteriors, and then the unkotting tunnels are the standard ones.
Third, D could be a nonseparating compressing disk forW. Then the complement
of D in W is a solid torus. When N(a) is added to D, we obtain a nonseparating
annulus in M that cuts M into a solid torus. The only knot or link exteriors in S3
that are obtained by gluing together two parallel annuli on the boundary of a solid
torus are cable spaces, which have more than one boundary component, a contradic-
tion to M being a knot exterior.
The fact that the single maximal cusp must have a longitude of length less than 4
follows immediately from Theorem 1.1(ii). This completes the proof. &
THEOREM 1.5. Let G be a properly embedded orientable incompressible boundary-
incompressible surface in a connected orientable 3-manifold W, Let G0 be obtained by
tubing together pairs of boundary components of G. If D3W is hyperbolic and G0 is
compressible, then there must be a longitude with length less than 4 in the cusp
boundary in any set of cusps with disjoint interiors in one of the components of D3W.
Proof. The copies of the surface G becomes totally geodesic in D3W, and the
existence of a compressing disk in D3W for G0 causes the existence of a closed path
through an alternating sequence of geodesic planes and horospheres that bounds a
disk in H3. As in the proof of Theorem 1.1, this forces the existence of a longitude of
length less than 4. &
Acknowledgements
Thanks to Abigail Thompson and William Menasco for helpful conversations. The
work in this paper would not have been possible without the explicit examples that
one can compute using the SNAPPEA hyperbolic structures program, written by
Jeffrey Weeks. It is available at http://humber.northnet.org/weeks/. This work was
supported in part by NSF Grant DMS-9803362.
References
1. Adams, C.: The noncompact hyperbolic 3-manifold of minimal volume, Proc. Amer.Math. Soc. 100 (1987), 601–606.
2. Adams, C.: Unknotting tunnels in hyperbolic 3-manifolds, Math. Ann. 302 (1995),177–195.
3. Adams, C.: Maximal cusps, collars and systoles for hyperbolic surfaces, Indiana Univ.Math. J. 47(2) (1998), 419–437.
4. Adams, C. and Reid, R.: Unknotting tunnels in two-bridge knot and link complements,Comment. Math. Helv. 71 (1996), 617–627.
5. Boileau, M., Rost, M. and Zieschang, H.: On Heegaard decompositions of torus exteriors
and related Seifert spaces, Math. Ann. 279 (1988), 553–581.6. Eudave Muoz, M.: On nonsimple 3-manifolds and 2-handle addition, Topology Appl.
55(2) (1994), 131–152.
DETECTING INCOMPRESSIBILITY OF BOUNDARY IN 3-MANIFOLDS 59
7. Eudave Muoz, M. and Uchida, Y.: Non-simple links with tunnel number one, Proc. Amer.
Math. Soc. 124(5) (1996), 1567–1575.8. Finkelstein, E. and Moriah, Y.: Tubed incompressible surfaces in knot and link comple-ments, Topology Appl. 96(2) (1999), 153–170.
9. Kobayashi, T.: Classification of unknotting tunnels for two bridge knots, In: Proc.Kirbyfest (Berkeley, CA, 1998), Geom. Topol. Monogr. 2, Geom. Topol., Coventry,1999, pp. 259–290.
10. Kuhn, M.: Tunnels of 2-bridge links, J. Knot Theory Ramification 5(2) (1996), 167–171.11. Menasco, W.: Closed incompressible surfaces in alternating knot and link complements,
Topology 23(1) (1984), 37–44.12. Morimoto, K. and Sakuma, M.: On unknotting tunnels for knots,Math. Ann. 289 (1991),
143–167.13. Morimoto, K., Sakuma, M. and Yokata, Y.: Identifying tunnel number one knots,
J. Math. Soc. Japan 48(4) (1996), 667–687.
14. Norwood, F. H.: Every two-generator knot is prime, Proc. Amer. Math. Soc. 86(1) (1982),143–147.
15. Oertel, U.: Closed incompressible surfaces in complements of star links, Pacific J. Math.
111(1) (1984), 209–230.16. Sakuma, M.: The topology, geometry and algebra of unknotting tunnels, Chaos, Solitons
and Fractals 9(4/5) (1998), 739–748.17. Scharlemann, M. G.: Tunnel number one knots satisfy the Poenaru conjecture, Topology
Appl. 18(2–3) (1984), 235–258.18. Weeks, J.: The SNAPPEA hyperbolic structures program, available at http://humber.
northnet.org/weeks/.
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