Topology Vol
Prmed
24. No
4. pp. 499-504.
Gw-9383185
1985
c
m Great Bntam
1985. Pcrgamon
13.00+
.a0
Press Ltd.
zyxwvutsrqponm
ON KNOTS THAT ARE UNIVERSAL
HUGH M. HILDEN, MARi. TERESALOZANO+
zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH
and JOSE MARia MONTESINOS.~
(Received
15June 1984)
41. INTRODUCIION
A link or knot zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
L in S3 is called universal if every closed, orientable 3-manifold can be
represented as a covering of S3 branched over L. W. Thurston introduced this concept in his
paper [6], where he also exhibited some universal links, and asked if the “figure-eight knot”
was universal. The question for the trefoil knot, as well as for any torus knot or, more
generally, for iterated torus knots or links, has negative answer, because these links, being
fibers of a graph-manifold structure of S3, can only be the branching set of graph-manifolds
(compare PI).
In this paper we answer the question of Thurston in the affirmative and we prove that
every non toroidal 2-bridge knot or link is universal. The proof uses the fact that the
Borromean rings are universal [43.
52. DIHEDRAL COVERINGS
zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
BRANCHED
OVER RATIONAL LINKS
To each rational number a/b there is associated the 2-bridge link L(a/b) shown in
Figure 1. In (a), the central tangle consists of lines of slope + b/u, which are drawn on a square
“pillowcase” (compare [S], [3]). In (b) this link is isotoped to exhibit the two bridges. The link
L(a/b) has one component if and only if a is odd. If L(a/b), L(a/b’) are such that b = b’ (mod
a) then they are equal and if b E -b’ (mod a) then L(u/b) is the mirror image of L(u/b’). Thus
we assume that a and b are relatively prime and that a > b. The link L(u/b) is toroidal if and
only if b = f 1 (mod a).
According to [l, p. 1613 z1 (S3 - L(u/b)) admits a representation f onto the dihedral
group Dzo of 2u elements such that the image of a meridian is sent to a reflection. The group
Dllr has the presentation {x, y : x2 = y2 = (x y)” = 11. Let a be the transitive immersion of D2,,
into the symmetric group S, of a indices defined by
a(x) = (12)(34)(56). . .
a(y) = (23)(45)(67) . . .
We now describe the u-fold irregular dihedral covering of S3 branched over L(u/b) associated
to the representation of: x1 (S3 - L(u/b)) - + S., where we can assume x, y are the meridians
shown in Fig. l(b).
Fig. 1.
+Supportedby “ComisibnAsesora
Figure-eightknot L(5/2).
de lnvestigacibn
Cientifica
499
y T&mica”.
500
Hugh M. Hilden. Maria Teresa Lozano and Jose Maria Montesinos
There is a sphere S2 dividing S3 in two balls A and B, such that (A, L(ajb) zyxwvutsrqponmlkjihgfed
n A) and
(B, L(a/b) n B) are homeomorphic to a 3-ball with two properly embedded unknotted and
unlinked arcs as the one shown in Fig. 2. We can imagine St as a small “pillowcase” parallel to
the one onto which lie the tangled part of L(a/b).
Thus the covering of S3 branched onto L(a/b) with monodromy zf is divided by the
preimage of S2 in two parts d and B, covering A and B respectively. Both 2 and B are 3-balls
and thus the covering we are describing is just S3. The covering A -, A for the knot L(5/2) is
shown in Fig. 3, where we have also depicted the intersection of the preimage of Dr u D2 c B
with 8A. The preimage of the branching set in the covering S3 -, S3 branched over L(a/b) is
the union of the preimages of L(a/b) n A and L&/b) n B. The preimage of L(a/b) n A are the
a + 1 arcs properly imbedded in A’shown in Fig. 3. The preimage of L(a/b) zyxwvutsrqponmlkjihgfedcbaZYXW
n B can be isotoped
through the preimage of Dr u D2 c B into a,& so that we can think of the preimage of
L(a/b) A B as a subset of the arcs lying on 8A. Namely, if we delete just one of the two arcs
that coalesce in the same point of the preimage of L(a/b) n A, for each such a point, we obtain
the preimage of L(a/b) n B, up to isotopy in B. In Fig. 4 we have depicted the preimage of the
branching set corresponding to the knot L(5/2). Figure 4(b) shows the branching cover after
puncturing the “pillowcase”, twisting one of its ends and flattening out onto the plane.
Similarly, Fig. 5 shows the branching cover corresponding to the link L( 12/5) under a
dihedral covering of 6 sheets, obtained by an analogous procedure. This cover will be used
later.
The preimage of the branching set of the figure eight knot L(5/2) is the link of Fig. 6. This
is an amphicheiral link to which we will refer as the “roman link”+.
Fig. 2.
Fig. 3.
(a)
(b)
Fig. 4.
7 This link appears
as part of the decoration
of a roman
mosaic zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO
fo und in the city of Zaragoza.
501
ON KNOTS THAT ARE UNIVERSAL zyxwvutsrqponmlkjihgfedcbaZYXWVU
Fig. 5.
Fig. 6.
$3. THE ROMAN LINK IS UNIVERSAL
THEOREM 1. There is a covering p:S3 -+ S3 branched over the roman link such that the
preimage of the branching set contains the Borromean rings. Thus the roman link is universal.
Proof. The roman link of Fig. 6 is depicted in Fig. 7 with an assignment of permutations
to the components A and B. The corresponding dihedral covering of 4 sheets is described in
Fig. 8(a). Figure 8(b) shows the preimage of the component C. This preimage is the rational
link L(12/5). In the dihedral cover of 6 sheets branched over L(12/5), shown in Fig. 5, we
extract the link Co u C2 u C5 u C6 which we depict in Fig. 9. The preimage of Co u Cg under
the dihedral covering of 3 sheets branched along C2 u CS is shown in Fig. 10. The preimage of
B under the 3-fold cyclic covering of S3 branched over A are the Borromean rings (Fig. 11).
Now in [43 it was shown that the Borromean rings are universal. This finishes the proof of
the Theorem.
COROLLARY
2. The Figure-eight knot is universal.
Fig. 7
Hugh M. Hilden. Maria Teresa Lozano and Jose Maria Montesinos zyxwvutsrqponmlkjihgfedc
502
(b)
Fig. 8.
$g$
/
C,!
Acc
F
/
c,
C6
Cl---
fC2
1
!
!
cc
Fig. 9.
Fig. 10.
Fig. 11.
$4. UNIVERSALITY
OF RATIONAL
KNOTS
AND
LINKS
THEOREM 3. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
Thepreimage of L(a/b) under thea- fold covering S3 + S3 branched over L(a/b)
and withmonodromy group the dihedral group of 2a elements D2,,, contains the “roman
subiink if’and only $ b f + 1 (mod a), i.e. ij L(a/b) is not toroidal.
Proof. We first assume
the a-fold
p-l
covering
(L(a/b))
has
b c t. Figure
p: S3 - B S 3 that
1+
[I
4
components
12 shows the preimage
we are considering
link” as a
of L(a/b) (= L(8/3)) under
(compare
with Fig. 4). The link
C,, . . _ , CL,:21, and we claim that the link zyxwvutsrqponmlk
C,,, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG
if b # 1, is the roman link co u c, u c, of Fig. 4(b) and 6. It is evident
that Co u Cb is equal to co u c,. Now the partner of c, in Fig. 7 must be a C, such that
0 < x < b and with the additional property that traveling down along C,, starting from the
CO
u
cb
u
co
-
[u/ l,]
b 9
503
ON KNOTS THAT ARE UNIVERSAL
Ptllowcase
Fig. 12.
left side of the pillowcase (see Fig. 12) we reach the first bridge of C, before touching the
middle line of the pillowcase. This guarantees that C, lies on the plane (i.e. its projection has no
%b
[I
; ,b
([I
>
satisfies
double points), and behaves exactly like the curve c, of Fig. 4(b). Now x0 = a - zyxwvutsrqponmlkjihgfedcbaZ
0 < x0 < zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
b and the first bridge of CxOis reached in the point of coordinates(X,Y) =
with respect to the reference shown in Fig. 12. Since
of the pillowcase, the component
%, >
([I
b
lies to the left of the middle line
C,, together with Co and Cb is the roman link.
If b > i the link L % is the mirror image of L (&).Thusp-‘(L($)containsthe
0
mirror image of the roman link, i.e. the roman link.
Finally, since L(a/l) is not universal, p-l (L(a/l)) can not contain the roman link as a
sublink. This finishes the proof of the Theorem.
COROLLARY 4. Eoery
Remark.
rational knot or link which is not toroidal is universal.
Using similar methods to that of [4] we can prove that many 3-bridge knots are
universal.
Question. Is every hy perbolic knor
universal?
REFERENCES
1. R. H. Fox: A quick trip through knot theory. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE
Topology of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC
3-mangolds
and Related Topics. Prentice-Hall:
Englewood Cliffs (1962).
2. C. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
MCA. GORDON and W. HEIL: Simply connected branched coverings of S’. Proc. Am. Math. Sot. 35 (1972),
287-288.
3. A. HATCHER and W. THURSTON: Incompressible surfaces in 2-bridge knot complements. fnuent. Math. (to
appear).
504
Hugh M. Hilden, Maria Teresa Lozano and Jo& Maria Montesinos zyxwvutsrqponmlkjihgfedcbaZYXWV
4. H. M. HILDEN,M. T. LOZANOand J. M. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE
MONTESINOS:
The Whitehead link, the Borromean ringsand the knot gd6
are universal, Collectanecl Mathemtica
XXXIV
(1983), 19-28.
5. H. SCHUBERT:Knoten mit zwei Briicken. Math. Z. 65 (1956), 133-170.
6. W. THURSTON:Universal links. (preprint, 1982). zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJI
Facultad de Ciencias,
Universidad de Zaragoza,
Spain.
Department of M athematics,
University of Hawaii,
Honolulu, Hawaii 96822.
USA.
M athematical Sciences Research Institute,
1000 Centennial Drive,
Berkeley ,
California 94720,
USA.