Invent. math. 109,473-494 (1992)
/~r/ve~/t/ones
mathematicae
9 Springer-Verlag 1992
Cusp structures of alternating links
I.R. Aitchison, E. Lumsden, and J.H. Rubinstein
University of Melbourne, Department of Mathematics,Parkville, Victoria 3052, Australia
Oblatum 20-V-1991
Summary. An alternating link Lfr is canonically associated with every finite,
connected, planar graph F. The natural ideal polyhedral decomposition of the
complement of 5@ is investigated. Natural singular geometric structures exist on
S 3 - ~ r , with respect to which the geometry of the cusp has a shape reflecting the
combinatorics of the underlying link projection. For the class of'balanced graphs',
this induces a flat structure on peripheral tori modelled on the tessellation of the
plane by equilateral triangles. Examples of links containing immersed, closed
zl-injective surfaces in their complements are given. These surfaces persist after
'most' surgeries on the link, the resulting closed 3-manifolds consequently being
determined by their fundamental groups.
Introduction
One of the motivating examples for the study of geometric structures on 3manifolds is Thurston's description [Th] of the complement of the figure-8 knot.
This knot admits an alternating projection, from which a decomposition of the
complement in S 3 can be determined, as the union of two ideal tetrahedra with
faces identified. The procedure generalizes to alternating, prime link complements,
and has been investigated further by Menasco I-Mel, 2], Lawson [La], Takahashi
[Ta] and Weeks [Wel].
The graph obtained by projection of such links completely determines the link,
up to reflection in the projection plane. Face identifications of the two corresponding polyhedra, arising from the upper and lower hemispheres of S 3 determined by
the projection plane, can be read directly from the graph.
In the past, interest has focussed on finding explicit shapes for ideal tetrahedra
further decomposing these two polyhedra, so as to endow the link complement
with a complete metric of constant curvature - 1. For general alternating links,
such triangulations are not canonical, even though the resulting complete structure
is unique. Hatcher [Ha] has used the canonical polyhedral structure in the
identification of certain arithmetic links complements. These complements have
'balanced' diagrams (defined later).
474
I.R. Aitchison et al.
Another line of research has been to determine the existence of incompressible
surfaces in the complements of such links. Thurston [ T h ] showed that the complement of the figure-8 knot contains no non-peripheral incompressible surfaces,
and that all but finitely many Dehn surgeries on the figure-8 knot produce closed
3-manifolds which are non-Haken and hyperbolic.
For irreducible Haken manifolds, Waldhausen [Wa] has shown that the
homeomorphy type is determined by the fundamental group. There is no currently
known analogous result for either non-Haken or hyperbolic manifolds. However,
Hass and Scott [HS] have recently proved the beautiful result that an irreducible
closed 3-manifold admitting a ~l-injective surface, satisfying the l-line, 4-plane
conditions, is similarly determined by its fundamental group.
We show that the polyhedral decomposition of a natural class of alternating
links, further canonically decomposed into geometric cubes rather than tetrahedra,
produces canonical 7za-injective surfaces in the link complement satisfying the
Hass-Scott conditions. By analyzing the geometric structure of the cusps, we will
show that these properties of such surfaces survive 'most' Dehn surgeries. Accordingly, the closed 3-manifolds obtained by such Dehn surgeries are determined by
their fundamental groups. We thus produce infinitely many explicit examples of the
phenomena described in JAR4].
Alternating links from 4-valent graphs
Let ~ denote an arbitrary connected finite planar graph. We consider f# embedded
in the plane R z c S 2 ~ S 3. fq consists of a finite set ~/~ of vertices, and a finite set
of edges, each with interior embedded in the plane. A deletion of an edge
corresponds to the deletion of the interior of the corresponding arc in the plane.
The medial graph 0/r is obtained by taking one vertex vl for each edge ei of if,
chosen at an interior point of the edge, and connecting two vertices vl, v2 by an
edge for each vertex of f# at which el and e2 are incident and adjacent. Alternatively, ~'~e is obtained by inscribing the dual polygon for each polygonal region of
the 2-sphere complementary to N, as in Fig. 1.
Jr is a 4-valent planar graph, and its complementary regions in the plane can
be 2-coloured, checker-board fashion. We choose white for the unbounded region
/
G
/,
MG"
Fig. 1
Cusp structures of alternating links
475
Fig. 2
of the plane, and grey for the other coloured regions. (Fig. 2.) To white regions we
associate a ' + ' sign, and a ' - ' sign to the grey. Equivalently, the regions of
R 2 - - ~//{~ coloured grey are those containing a vertex of ~.
Each vertex of ~r can be converted to a crossing as in Fig. 2, so as to produce
a planar representation of an alternating link 2,o. Crossings viewed from the
interior of any region have sign that of the region. We will assume from now on
that the graph ~ contains no loops, and no edge-deletion disconnects .% The
corresponding planar projection of Ae~ is then reduced. Equivalently, regions
meeting at any vertex of 5~162are all distinct. Menasco has proved that if the
diagram is connected, the link is unsplittable [Me2].
Polyhedral face identification rules
The projection plane separates S 3 into two balls, the upper and lower hemisphere.
There is a natural way to obtain the complement of the link as the union of two
ideal polyhedra with faces identified. This procedure was introduced by Thurston
[Th], and has been elaborated upon by Hatcher [Ha], Lawson [La], Menasco
[Mel], Takahashi [Ta] and Weeks [ W e l ] . Following Takahashi, we recall rules
for producing the polyhedra and face identifications as follows:
1. The graph J/4'~ on S 2 c S 3 endows each of the 3-ball hemispheres ofS 3 with the
structure of an (abstract) polyhedron. Let F~ (resp. F.~ ) denote the polyhedron
above (resp. below) the projection plane, as viewed by the reader. Note that
reflection in the plane identifies these two polyhedra, and this describes S 3 as the
union of F ~ , with corresponding faces glued by the identity map.
2. Label corresponding vertices of F~ and F~ using symbols vi, gi3. At each vertex vi of F~, label a short arc parallel to the over-arc at vi by symbols
x~, yz, x~ as indicated in Fig. 3. Thus y~ is a new label at the vertex, and the labels
x~ provide a label for each edge of the link projection.
4. Similarly label F~, labelling a short arc parallel to the under-arc at g~ by
symbols y~, x~, y~.
5. The short arcs of each graph separate the four corners of the regions meeting at
each vertex into two classes. Label each corner at v~, ~i by the symbols r~s~ where
476
I.R. Aitchison et al.
X~
Fig. 3
~4
X4
Xx ~
xN~~
x~y,
__
Ys
Fig. 4
rl = xi or Yl corresponds to the label at the vertex, and s; = yj or xj is the label on
an edge of the region at'the vertex not corresponding to a short arc. (Fig. 4.)
6. Carry out face identifications of corresponding faces so that corners labelled
Xiyk, ykXi are identified.
7. After deleting all vertices, denote the resulting 3-manifold by .~}.
The following result is essentially in the papers cited above. An alternative
description of this result appears in [AR3].
Theorem. The manifolds S a - 59~ and X } are canonically homeomorphic.
For each planar region R complementary to J/g~, denote by R + the corresponding faces of F ~ , and let e(R) and 6(R) denote the corresponding sign and
degree of R as a signed polygon.
Lemma. I f e(R) = + and 6(R) = n, the corresponding identification R+ ~ R_ is
achieved by 2z~/n clockwise rotation. I f e(R ) = - , the rotation is anti-clockwise.
Under these identifications of compact faces, we obtain a topological space with
singularities, each of which being a cone on a 2-dimensional torus. Deleting
vertices, we recover the complement of the link in S 3. The structure obtained is
exactly that described by Thurston [Th] for the Borromean rings. This example,
together with the figure-8 knot and Whitehead link, are the prototypes of the
construction, illustrating how alternating link complements are constructed by
glueing together two ideal polyhedra, face identifications 'alternating like gears'
[Th]. It would appear that Thurston is aware of many of the constructions of this
paper, despite their non-explicit appearance in [Th].
As a consequence, all face identifications can be read off directly from the
alternating projection. Immediate is the determination of equivalence classes of
edges under identification of polyhedral faces.
Corollary. Every edge equivalence class contains four edges.
Cusp structures of alternating links
477
Proof. Two corresponding vertices of F ~ give rise to a distinguished set of four
equivalent edges, as in Fig. 5, being those on the left (resp. right) as the vertex is
viewed from a ' +'-labelled region of Fff (resp. F~ ). An equivalence class of edges is
a minimal invariant set under face identifications. Such sets from a given crossing
point are minimal and invariant.
The combinatorial structure of cusps is now clear. The vertices of squares are
identified in sets of 4, by the previous corollary. With the link projection plane in
R 3 taken as horizontal, the edges of 3f~ appear as vertical lines running between
the over and under arcs, one such for each crossing point. The polyhedra being
identified have all vertices of degree 4, and their truncations give rise to squares at
vertices. These squares fit together to form the boundary tori of neighbourhoods of
the link components. Each such square is wrapped about some torus, with a pair of
opposite vertices identified on one of these vertical lines at crossings, as in Fig. 6
[Ta, Wel]. One diagonal of each square produces a meridian for a link component.
We endow each combinatorial cusp with a flat geometry, by declaring each square
to be a standard unit square.
Corollary. The combinatorial structure of cusps is canonically flat, bein9 locally the
standard tessellation of the plane by squares.
We now identify the combinatorial shape of each cusp. Each link component
Li has 2nl crossing points, counting each self-crossing twice, for some integer nl.
Correspondingly there are 2n~ vertices on the two polyhedra F ~ which are
equivalent after face identifications. That one of the diagonals of each square
corresponds to the meridian of the cusp in question can also be seen by branched
covering S 3 along a small unknotted circle in the link complement linking L~ once,
to obtain a new alternating link. The longitude of Li lifts non-trivially, and the
meridian trivially. But the lifted cusp structures obey the same rules.
Fig. 5
Fig. 6
478
I.R. Aitchison et al.
The other diagonals of squares fit together on each torus to form two disjoint
closed circles, isotopic to a planar parallel of the corresponding link component.
Thus
Proposition. The geometry of a cuspidal torus whose combinatorial structure is
produced by 2n squares is determined by two orthogonal covering transformations zu,
z, of the Euclidean plane, corresponding to the meridian and planar parallel kt and p,.
These translations are by distances x//2 and nxf2 respectively. (Fig. 7.)
Definition. A torus built in this way from 2n squares will be said to have shape n.
Observe that the squares of the universal cover of each cusp can be coloured
like a checker-board, according to which polyhedron they arise from. The way in
which edges of these squares are identified can be read from the link diagram in
a very simple manner. The result is amply illustrated, although not explicit,
in Thurston's notes [Th]. Label the edges of all components of the link projection
J/g~ with symbols el, where it is convenient to have indices (cyclically) monotonic as
we move around each projected component, with respect to some arbitrarily
chosen orientation and initial edge.
Labels for the edges of the polyhedra F~ and F~ are obtained by replacing ei by
x~ and Yi respectively. After truncation of all 4-valent vertices, the vertices of the
resulting square tips can be labelled with the corresponding polyhedral edge label
(Fig. 8). Observe that in Fig. 8, the labelling of the corners of squares proceeds
mechanically. From F~, we take all squares corresponding to undercrossings of the
component L j, whereas from F~, we take squares corresponding to overcrossings.
"~I~
~~ ,
9
\/
Fig. 7
,4
4+118272+11484+iiii iii
Fig. 8
Cusp structures of alternating links
479
These squares can be labelled with a ' + ' and a ' - ' respectively, and with corner
indices arranged so that those labels corresponding to edges of the undercrossing
arc occur in the bottom left and top right positions when the squares are identified
with the standard unit square in the plane. This is illustrated in Fig. 8, where
squares coming from F~ have corners labelled in the same cyclic order as that of
the view of the plane of projection, whereas those from F~ have order reversed.
Hence reading from left to right, the labels in the top left of squares follows the
order in which they appear as we move around the projected component. The same
is true for those labels appearing in the bottom right positions, since the top left and
bottom right are consecutive labels, consistent with the labelling of a parallel of the
knot.
Surfaces from canonically cubed generic polyhedra
We have remarked that there is no canonical decomposition of an ideal polyhedron
into ideal tetrahedra.
Definition. A polyhedron is generic if each vertex is of degree 3. This terminology
arises from the polyhedra obtained by intersecting half-spaces in R a defined by
planes in general position.
An exceptionally convenient circumstance prevails for generic polyhedra, since
we obtain a canonical decomposition into 3-dimensional cubes. Such a decomposition arises by introducing a new vertex for each edge, face, and the center of the
polyhedron. Join each new face-vertex to adjacent edge-vertices, thereby creating
squares around the original vertices, three at each, which cover the 2-sphere. The
cone on three squares is combinatorially a cube, giving a decomposition of the
polyhedron into cubes, one for each original vertex. (Fig. 9.) In the compact case,
we may impose a geometry by declaring each of the cubes to be a standard
Euclidean unit cube. Each of these cubes can be further subdivided into 8 sub-cubes
by slicing with three squares parallel to the three pairs of opposite faces. For
a closed 3-manifold built from trivalent polyhedra, after subdivision into cubes,
these new squares fit together to define a possibly disconnected and non-orientable
immersed surface canonically associated with the combinatorial structure ([AR 1]).
We will carry out a similar construction for a specific class of non-compact
alternating link complements, using the ideal polyhedral structure described above.
Fig. 9
480
I.R. Aitchison et al.
Collapsing bigons: balanced alternating links
"Let bigons be bygone" (W. Menasco)
The polyhedra obtained above from an alternating link projection are 4-valent. In
certain circumstances, we can obtain the same link complement by identifying faces
of two trivalent ideal polyhedra, as occurs with Thurston's description of the
figure-eight knot. See also Hatcher [Hal.
Suppose we obtain a bigon in the link projection. The opposite edges of the
bigon fall into distinct edge-identification classes. The two discs in F~ and
F~ defined by the corresponding bigons are identified as an embedded disc in the
quotient, which can be collapsed to become a single edge. The effect on the
polyhedral identifications is to replace each of the two bigons with a single edge,
the two degree 4 vertices of each bigon becoming degree 3, as in Fig. 10. We will
denote by H.~ and H~ the polyhedra resulting after collapsing possible bigons.
Remark. The order of collapse is not canonical. The only circumstance in which
a bigon cannot be collapsed in this way is if the two edges are in the same
equivalence class (Fig. 11), as can occur with the Hopf link. The problem arises only
after some initial allowable collapse of a bigon.
Lemma. Any chosen bigon of F~.~ and F~ can always be collapsed.
Hence the quote due to Menasco [Me1].
If the link projection has several bigons, after allowable collapses the resulting
polyhedra may have a mixture of vertices of degree 2, 3 and 4. Consequently cusp
structures will involve, at the combinatorial level, bigons, triangles and squares
glued together.
Definition. We say that a reduced alternating link projection has balanced bigons,
or is balanced, if every crossing point is a vertex of exactly one bigon.
/-\
/_\
Fig. 10
Fig. 11
Cusp structures of alternating links
481
In this situation, there is no ambiguity in the order of collapse, and every
possible collapse is allowable./L~ a n d / / ~ are generic ideal potyhedra. From now
on we will consider only balanced, reduced, alternating link projections. A link
admitting such a projection will be called balanced.
Lemma. Faces of H~ and II#~ are naturally 2-coloured, with both colours occurring
at each vertex. Edge identification classes contain six edges.
Proof. Since there is exactly one bigon at each vertex of H ~ , and faces are
checkerboard coloured, both colours still occur after collapsing bigons. There are
two equivalence classes of four edges each corresponding to the two vertices of
a given bigon. Two of these edges are identified in each of the two polyhedra,
resulting in six edges in the equivalence class (Fig. 12). This can also be seen directly
from face identifications.
The ideal polyhedra H~{ can each be decomposed into cubes, one for each ideal
vertex, with each cube similarly having a vertex deleted. Each cube has three types
of edges: those from the original edges of/-L+ , those lying in faces o f H ~ , and those
from the centres of faces to the centres of the polyhedra. After identifying faces to
produce the link complement, edges of the first two types are common to respectively 6 and 4 cubes, whereas an edge of the latter type corresponding to an n-gon
face is common to n cubes.
We describe a canonical model for such structure. Consider the tessellation
{4, 3~ 6} of hyperbolic 3-space by regular ideal cubes, with dihedral angle 2~/6
[Col, 2]. Symmetrically decompose one such cube into 8 subcubes, each with one
ideal vertex and all dihedral angles either ~/2 or ~/3. We endow each cube in the
decomposition of H ~ with geometry modelled on such a subcube.
Definition. Call a balanced projection nicely-balanced if there are no triangular
regions in the projection plane. Denote by ~ the class of prime links admitting
nicely-balanced projections.
Theorem. Suppose Lf ~#/" is a nicely-balanced prime link. Then S 3 - ~
admits
a complete, possibly singular metric of curvature < - 1. The sin#ularities of the
metric are concentrated along a connected 9raph with two vertices and all edges
of equal length, contained in a compact set of diameter bounded by a universal
constant C.
Proof. Choose a nicely-balanced projection for S . With each polyhedron
H ~ decomposed into hyperbolic cubes as above, the geometry of the link complement is as claimed in the complement of the l-skeleton. All edges of the original
polyhedra have dihedral angle ~/3, and accordingly the constant curvature
\/
Fig. 12
482
1.R. Aitchison et al.
geometry extends across such edges, as they fall into equivalence classes of 6. All
other edges have cone angle k ' ~ / 2 for some k > 4. Vertices at the centre of an
n-gon face have link a 2-sphere triangulated as the double suspension of an n-gon,
each triangle endowed with a constant curvature + 1 metric, all angles 7t/2.
Vertices at edges are non-singular. Finally, the two vertices at the centres of each
polyhedron have identical link type, the triangulation of S 2 dual to the trivalent
graph defining H • Each triangle is again right-angled spherical, and since ~ is
prime, all closed geodesics in this spherical metric have length > 2~. (For further
details, see JAR1].)
The singularities of the metric are thus concentrated along a subset of the graph
with vertices at the centres of each polyhedron, and with edges joining these,
passing through the centres of faces of the original polyhedra. All of these edges
have the same length, and so the graph is contained in a compact set of universally
bounded diameter.
We recall that a stronger result can be obtained:
Theorem [Me2 ] Every non-splittable prime alternating link which is not a (2, q)torus link has only peripheral ~l-injective tori.
Invoking Thurston's theorem on the existence of hyperbolic structure,
Corollary [Me2] Every non:splittable alternating prime link which is not a (2, q)torus link has a complete metric of constant curvature - 1.
Our combinatorial approach has significant advantages: Using the model
described above for the compact case, we construct a canonical surface immersed
in the link complement, associated to the combinatorial decomposition. Each cube
has six faces, three compact, three non-compact. Given e > 0, take three compact
'squares' in each cube, each the boundary of an e-neighbourhood of a compact face
of the cube. These squares fit together to produce a compact immersed surface,
possibly disconnected and non-orientable, in S 3 - ~ . The family of surfaces
~-~ obtained by varying e can be viewed as the images under regular homotopy of
an immersed surface ~ . Letting e become zero, Y collapses onto the 2-complex of
compact squares as a totally geodesic surface with respect to the singular metric.
Accordingly, each pre-image of ~ in the universal cover of S 3 - S is a plane, for
e > 0 sufficiently small, any two distinct translates of which being either disjoint or
meeting along a line close to being geodesic. Similarly, the angle of intersection of
two sheets can be presumed arbitrarily close to a right-angle. Hence
Theorem. Each link complement S 3 - ~ , for L#6~tt/~, contains an immersed ~linjective closed surface, satisfying the 4-plane, l-line condition of Hass and Scott
[US].
Cusp geometry of nicely-balanced links
For ~ e ~ ,
a pair of corresponding bigons in the polyhedra F ~ contributes
4 ideal vertices, and thus 4 squares to peripheral tori after vertex truncation. Each
such square has a distinguished edge arising from the bigon. Consequently, as the
bigons are collapsed to produce H f , each square of a peripheral torus is collapsed
to a triangle. This provides a convenient means by which to identify generating
Cusp structures of alternating links
483
covering transformations for each torus. We know that after identifying faces of
/ / ~ , edges fall into equivalence classes of six, giving a combinatorial decomposition of each peripheral torus into triangles, with universal cover combinatorially
the standard lattice of equilateral triangles. This tessellation of the plane arises by
taking the standard rectangular lattice, and collapsing to a point one edge for each
square. Such a possible collapse is illustrated in Fig. 13.
The 2n squares of a torus of shape n become 2n equilateral triangles, and
accordingly we continue to say such a triangulated torus has shape n. Each vertex
of the tiling of the plane by squares has a unique incident edge which is contracted,
and each square has exactly one edge to be collapsed. We have then a 'perfect
matching' on the square lattice, with the additional properties of periodicity and
only one distinguished edge per square and vertex.
One of the non-horizontal directions in the tessellation {3, 6} corresponds to
the meridians for cuspidal tori. With cusp geometry determined by the tips of ideal
regular cubes, generators for covering transformations t, and t,, for the cusps are
given respectively by translations of the complex plane by complex numbers
z, = (1/2 + x/-3/2i)2
z, = n(1/2 + xf3/2i)2,
with z u corresponding to the meridian, up to an arbitrary scaling constant 2.
A more careful analysis enables us to identify the image of a planar parallel of
a link component as a vector in the triangular lattice. The following argument is
due in part to Lawrence Reeves, for which the authors are thankful:
Observe (as in Fig. 8) that positive-crossing bigons contribute horizontal edges
of squares to be collapsed, whereas negative-crossing bigons contribute vertical
\
\
t/////l//l
Fig. 13
484
I.R. Aitchison et al.
edges. Moreover, the squares in the plane are of 2 types, c010ured (signed) according to the polyhedron from which they arise. Hence the planar parallel 7, of Fig. 7
passes through n squares of the same type (corresponding to one of the two isotopic
copies on the cusp torus in question lying above and below the link component).
Consider two squares $2, $4 in the plane, with common vertex the origin, and
lying in the second and fourth quadrants respectively. Consider the edges e2, e~ of
these squares marked for collapse. We will use the symbols t, b, s and r to denote
the top, bottom, left and right edges of a square. Denoting the marking of all
squares symbolically by M, we see M(SI)E {t, b, s, r}.
Since each vertex must be incident to a marked edge, we find:
If M ( S 2 ) --- t, then
If M ( S 2 ) = s, then
If M(S2) = b, then
If M(S2) = r, then
M(S4)
or M ( S 4 ) = s.
t or M ( S 4 ) = s.
b or M(S4) = r.
b or M ( S 4 ) = r.
= t
M(S4) =
M(S4) -
M(S~) =
Thus strips of squares parallel to 7, involve only squares labelled with symbols
in {s, t}, or in {r, b}. Note that these strips alternate, and that the meridianal
translation is a symmetry of the marking of squares. Hence vertical translation
upwards by one unit converts a bi-infinite sequence of s's and t's to a bi-infinite
sequence of r's and b's, with the correspondence interchanging the symbols s and r,
t and b. Translation downwards by one unit involves a shift-map, as does translation upwards by two or more units.
It is convenient to choose as origin a vertex for which the squares $2, $4 have
labels in {s, t}. Reading down to the right from the origin then gives an infinite
sequence of symbols obtained by infinitely repeating a word w of length n in
symbols s and t.
When each square is collapsed to a triangle, the distinguished diagonal corresponding to the meridian becomes a distinguished edge determining the orientation
of the triangle in the plane. We orient and label edges of triangles in Fig. 13 using
symbols tr, 7 and /~, with /~ pointing upwards, cr horizontal, and 7 pointing
downwards. All orientations point to the right.
The planar parallel 7, gives a path beginning at the origin, and passing
downwards through the diagonals of n squares. To each vertex of this path we
associate one of the ordered pairs of symbols tt, st, ts, or ss according to the
marking of the squares. After collapsing squares, each such vertex becomes the
centre of a configuration of six triangles, and the two corresponding diagonals of
squares become two distinct edges of this configuration passing through the centre.
The planar parallel 7,, originally described by the word w in symbols s and t,
becomes a word co in symbols 7, tr and #. The correspondence is obtained by
replacing s by a, and t by 7.
If the link component involves k negative bigons, the symbol s occurs k times in
w, where 0 _< k < n. As a vector in the plane, we find 7, becomes the vector
z * = (n - - k)~ + krr
= (n -
k)~ + k ( 7 +
= nr + k#
= z..4- kzu .
~)
Cusp structures of alternating links
485
Hence the covering transformation corresponding to the planar parallel is t.t k. It is
thus convenient, when dealing with a general link component, to use the more
canonical generators t, and t,. We will refer to the closed circle on a cuspidal torus
corresponding to t, as a k-parallel.
A regular ideal cube can be equivariantly truncated by expanding a horosphere
at each vertex, until the vertices of the sliced-off equilateral triangles meet. In this
fashion we obtain a canonical decomposition of the alternating link complement
into a compact 'thick' part, and a finite number of cusps constituting the 'thin' part
[Th]. The latter compactify naturally as T 2 • [1, z~), with each total slice having
flat geometry decaying exponentially as we move out to infinity. The length of the
meridian on T 2 x {1}, the boundary of the maximal cusp, is independent of n. In
the upper-half space model for hyperbolic 3-space, we may assume that the cusp is
the quotient of the region above the horizontal plane at height 1. This is the
convention used by Weeks in his 'snappea' programs [We2], and allows us to
assign specific lengths for the meridianal and longitudinal translations.
Lemma. The meridian on the boundary torus of a maximal cusp is of length x/~, and
so we may normalize by taking )~ = x/2 in the equations above.
Proof This can be calculated directly as an exercise in hyperbolic geometry, by
maximally truncating an ideal regular cube, and measuring the length of a side of
the resulting hyperbolic cuboctahedron. An 'easier' alternative is to utilize Weeks'
program to determine the cusp geometry of the alternating link 8 4 from Rolfsen's
tables. This balanced link has complement obtained by glueing together two
ideal regular cubes, and the toral geometry at height 1 has meridianal translation
length x/2.
Surgery and the 2n theorem
Since we have a clear picture of cusp geometry for all possible shapes arising from
the equilateral triangular lattice, we can completely characterize possible Dehn
surgeries on balanced links for which the geometry of the thick part of the link
complement extends over the surgery solid tori with non-positive curvature. Such
surgeries are those for which the attaching circle for a meridianal disc of the solid
torus is sufficiently long. For a cusp of shape n, the lengths of possible attaching
circles can be explicitly determined from the corresponding sublattice.
If 7 is a simple closed curve on the maximal torus T 2 • {1}, with length l~, the
length of the corresponding geodesic circle on T 2 • {t} is l~(t) = l~e -t. 7 is defined
by p l ~ - qp,, with respect to the meridian and planar parallel, for some p and
q relatively prime integers. Thus, after rotating by e i~/6
l~ = I(nq xfl3/ x/2, (2p -- nq)/ x ~ ) I .
The complex numbers z, and z, determine a sublattice F. of the triangular lattice,
and the excluded surgeries are those corresponding to the lattice points of F, lying
within the circle of radius 2n. For a nice exposition of Gromov-Thurston's 2n
theorem, we refer to Bleiler and Hodgson [BH].
We denote by k, + 1 the number of primitive lattice points of F, lying in the
half-disc {z: Izb < 2n} c~ {z: Re(ei~/6z) > 0}. Then k. is the number of excluded
486
I.R. Aitchison et al.
Fig. 14
surgeries on a cusp of shape n, since (p, q) and ( - p, - q) surgeries determine the
same manifold. N o t e that all points of F, lie on equally-spaced parallel lines, the
closest to the origin being at a distance d, = ]nqxfl3/x/2 I. There is only the one
excluded surgery _ (1, 0) on the line Re(ei~/6z) = 0, corresponding to the meridian
/~. (Fig. 14.)
Taking n > 6 and ]ql > 1 gives fnql > 6 and only the trivial surgery is excluded
since
d,, = L n q ~ / ~ l
> (7.34847)1q1 > 2re = 6.28318.
L e m m a . For a link 5r ~ ~U, no cusp has shape 1.
Proof A cusp has shape 1 if and only if it is built from two equilateral triangles. If
these arise from a self-crossing of a link component, that c o m p o n e n t is a figure-8 in
the projection plane disjoint from other link components. This contradicts the link
being non-splittable. The only other possibility is that the c o m p o n e n t in question
projects to an e m b e d d e d circle in the plane of the link diagram. If the two triangles
of the cusp arise from two intersections of some other c o m p o n e n t with this circle,
each intersection point is a vertex of two bigons, and the link diagram fails to be
balanced.
For n = 2, 3, 4, 5, we proceed on a case by case basis to calculate k,. We m a y
assume nq < 5, and n,q > 1. Hence q = 1 unless n = 2, for which q = 2 is also
Cusp structures of alternating links
487
Table 1
Shape
k,
Excluded Dehn surgery coefficients (p, q)
n=2
12
(I, 0), {1, 2), {3, 2), {p, 1) for -3__<p<5
n= 3
9
(1,0),(p, 1) for - 2 < p < 5
n=4
n= 5
6
3
(1,0), (p, 1) for 0 < p < 4
(1, 0), (2, 1), (3, l)
n> 6
1
(1,0)
possible. To exclude (p, q)-surgery requires
I ( n q x / 3 / ~ , (2p - nq)/x~)] < 2n
3nZq2/2 + (2p - nq)2/2 < 39.47835 .
The resulting possibilities are presented in Table 1, and for all other possibilities we
m a y conclude that a negative-curvature metric extends over the glued in solid
torus. The values of n for excluded lattice points are indicated in Fig. 14.
Definition. Suppose a = (al . . . . . at) is a sequence of Dehn surgery coefficients
assigned to the components of a link ~ . If every cusp of shape n is assigned
a surgery coefficient other than those k, possibilities listed above, we will call a an
allowable Dehn filling.
Theorem. Let S e ~ ,
and suppose a = (a~ . . . . . at) is an allowable Dehn.filling
defining the closed 3-manifold M ( ~ , a). Then M ( ~ , o-) admits a singular metric of
negative curvature.
Survival of ~l-injective surfaces after surgery
The metrics of negative curvature obtained in this way from nicely-balanced
alternating links contain isometric copies of the thick part of the corresponding
link complement. Accordingly the geometry of a n e i g h b o u r h o o d of the immersed
surfaces is preserved.
Theorem. If a is an allowable Dehn filling for 2 P ~ r ~, then M ( Y , a) contains
a 7q-injective surface satisfying the 4-plane, 1-line conditions of Hass and Scott.
Proof This is an immediate consequence of [HS], since the immersed surface of
the link complement persists as an essentially totally geodesic object, whose sheets
meet at an angle essentially n/2. By [HS], this guarantees the desired properties.
Corollary. If a is an allowable Dehn filling for ~ ~ ~~ then M(s a) has infinite
fundamental group, universal cover homeomorphic to R 3, and is determined up to
homeomorphy among closed, orientable, irreducible 3-manifolds by its fundamental
group.
That the universal cover of M ( ~ , a) is h o m e o m o r p h i c to R 3 is proved in
[ H R S ] using the existence of a nt-injective surface.
488
I.R. Aitchison et al.
Corollary. I f X is any knot in ~i4/"then every closed 3-manifold obtained by nontrivial Dehn filling on K contains a nl-injective surface, and is determined up to
homeomorphy among closed, orientable, irreducible 3-manifolds by its fundamental
group.
Remark. Such a knot always has an even number of crossings. The figure-8 knot is
the only non-trivial alternating knot with a smaller even number of crossings, and
although balanced, the faces of the corresponding polyhedra are triangles. Dehn
surgeries on the figure-eight knot are well-understood [Th], and the complement is
known to contain infinitely many immersed, closed totally geodesic surfaces [Re].
A more geometric understanding of such surfaces is clearly warranted.
Construction of examples
1. Thefigure-8 knot. This knot has balanced bigons, but the faces oftetrahedra are
of degree 3.
2. Borromean rings. The Borromean rings [ T h ] give an example where no bigons
occur.
3. Canonical constructions fi-orn marked planar graphs. For the constructions to
follow, we wish to obtain examples of 4-valent planar graphs giving rise to
non-split alternating links. Menasco [Me2] has shown that an alternating link is
non-splittable if and only if it has an alternating projection arising from a connected graph.
A projection of a balanced alternating link can be contracted to a planar
4-valent graph, with complementary regions 2-coloured checkerboard fashion, by
collapsing each bigon to a point. We may reverse this procedure. Conversely,
consider a planar 4-valent graph F, as in Fig. 15.
Definition. A marking of a vertex is defined by an additional arc, as in Fig. 16.
A state of F is an assignment of a marking to each vertex.
Such markings have occurred in the recent work of Kauffman and Murasugi on
Jones' polynomials of alternating links, as exposited for example in [ H K W ] .
Fig. 15
Fig. 16
Cusp structures of alternating links
489
A state of F can be given several interpretations, each of which enabling us to
associate a link to that state.
(i) Each marking determines a splitting, as in Fig. 17. Such splittings arise in the
work just mentioned.
(ii) Each marking defines a crossing, as in Fig. 18. This gives the encoding of all
possible link projections as the choice of graph varies.
(iii) Each marking defines a clasp. This could be chosen to always be positive or
negative, as in Fig. 19. The result would then not necessarily be an alternating
link.
(iv) Each marking defines a clasp, so chosen to make the resulting projection
alternating, according to the convention above.
(v) Take an arbitrary projection of a link onto the plane, and obtain a marked
graph as in (ii) above. Reinterpret this marking as in (iv). To obtain a balanced
projection, the original link projection has to be free of clasps.
In order to obtain a balanced projection from an arbitrary 4-valent planar
graph, we must preclude marking the vertices of any bigon of the projection as in
Fig. 20.
Remark. There is a notion of duality and complementation here, since there are
exactly two choices at each crossing. This manifests in different ways in different
mathematical contexts. (Fig. 21.)
Fig. 17
Fig. 18
Fig. 19
490
I.R. Aitchison et al.
Fig, 20
Dual itg
o f markers
Rotate by
~2
Tetrahedral
move
Feynman
d iggrams
Whitehead
moves
Rotat ion of
rooted tree
Fig. 21
Fig. 22
4. Nested circles and added arcs. Take an arbitrary collection of disjointly embedded circles in the plane, coloured black, and add an arbitrary number of disjointly
embedded red arcs with end-points on the black circles. Convert the resulting
graph to an alternating link as in Fig. 22. The advantage of this construction is that
the number of components of the resulting link can be specified beforehand.
Alternatively, the dual marking gives rise to a different link.
5. 2-bridge links and 3-symbol words. We can investigate those 2-bridge knots and
links arising by this construction. Take the group
= ( ~ , f l , y l ~ Z = 1, f12= 1,72= 1 , ~ y = ~ ) ,
and let toe ft. Associate to co a 2-bridge knot and 2-component link respectively
according to the prescription to Fig. 22 applied to the two graphs of Fig. 23. (This
may not give a reduced projection.) It is not difficult to read off from the word co the
size of each polygonal face of the ideal polyhedra so determined.
Cusp structures of alternating links
491
Fig. 23
6. 2-colouring states of a 3-valence planar connected graph. An alternative view
point is to consider a planar tri-valent graph which has edges 2-coloured, each
vertex having two black and one red edge. This can be reinterpreted in terms of
signs associated to the complementary regions, in which case every vertex is
surrounded by regions of different type, red edges separating regions of the same
sign. The black edges connect together to produce a number of disjointly embedded circles in the plane, each in correspondence with a component of the resulting
balanced link, as in Fig. 22 and 23.
7. Non-alternating links. Take cyclic covers over S 3, branched along some unknotted component. The link obtained in the cover will in general be non-alternating.
Iterating this procedure along different unknotted components produces very
complicated examples of non-alternating links with analogous cusp structure and
polyhedral decomposition.
8. Alternating links from polyhedra. The construction is reversible, in the sense
that given an abstract polyhedron defined by a trivalent 2-coloured graph as above,
we can associate an alternating link in S 3 whose associated polyhedral decomposition of the complement gives back two copies of the original polyhedron.
9. Truncated connected planar graphs. A simple class of such links arises as
follows. Let F denote an arbitrary connected planar graph, as in Fig. 24. Construct
the truncated dual as in Fig. 25. This has a canonical 2-colouring of the desired
kind, and gives rise to a balanced link as above. Such a link can be obtained directly
from the original polyhedron. All components are unknotted, and linking between
components is completely determined by the symmetric adjacency matrix A defining the graph, obtained by labelling the vertices arbitrarily and setting Ai~ -- 0, 1
according to whether vertices vl and vj are connected by an edge in the graph. This
facilitates computation of homology groups of 3-manifolds obtained by Dehn
surgery on the link.
We mention some constructions of generic polyhedra with no triangular faces.
Fig. 24
492
I.R. Aitchison et al.
Fig. 25
Definition. A generic polyhedron P defined by a connected planar graph F is
allowable if it has no monogons, bigons or triangles, and can be saturated by
disjointly embedded circuits, in the sense that every vertex lies on exactly one
circuit. (This is equivalent to the existence of a 2-colouring of edges or faces ~s
described above.) Moreover, the dual triangulation must satisfy the geometric
conditions described earlier, namely that each closed geodesic on the 2-sphere
obtained by declaring all triangles right-angled spherical with curvature + 1 has
length > 2n. Let d ~ denote the set of allowable polyhedra.
(a) Only the dodecahedron and cube are allowable Platonic solids.
(b) Given P ~ s ~ , denote by z*(P) the truncated dual. Then z * P ~ d ~ .
(c) Given P ~ d ~ , and an edge e of P, consider ae(P) obtained by slicing off the
edge e of P. This creates an additional rectangular region, and increases the size
of each polygon at the ends of e. Then ae(P)~ d ~ . It suffices to verify for the
two possible colourings of the edge e that the result can be 2-coloured. This is
illustrated in Fig. 26, and requires that if the edge is black, the red edges at both
ends must emerge on the same side.
(d) Given P e t i N , consider a(P) obtained by slicing off all edges of P simultaneously. This is illustrated in Fig. 27. Then ~ r ( P ) e d ~ .
It is clear that we can iterate these constructions, but it is not at all clear what
other relationships hold between the links constructed in any such iterated family.
not
Fig. 26
alloged
Cusp structures of alternating links
493
Fig. 27
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