289
Internat. J. Math. & Math. Sci.
VOL. 14 NO. 2 (1991) 289-292
GEOMETRIC PRESENTATIONS OF CLASSICAL KNOT GROUPS
JOHN ERBLAND
Department of Mathematics
University of Hartford
West Hartford, Connecticut 06117
and
MAURIClO GUTERRIEZ
Mathematics Department
Tufts University
Medford, Massachusetts 02155
(Received September 19, 1989)
The question addressed by thls paper is, how close is the tunnel number of
A
a knot to the minimum number of relators in a presentation of the knot group?
ABSTRACT.
(The
dubious, but useful conjecture, is that these two Invarlants are equal.
[I]) It has been
analogous assertion applied to 3-manifolds Is known to be false.
The
shown recently [2] that not all presentations of a knot group are "geometric".
maln result
that the tunnel number is equal to the nflnimum
in this paper asserts
number of relators among presentations satisfying a somewhat restrictive condition,
that Is, that such presentations are always geometric.
I.
DEFINITIONS AND CONVENTIONS.
X
CI(S 3
Let K be any non-trlvlal knot in
$3;
V
CI(N(K)
solid torus containing K;
V), the knot exterior.
Let G
the knot group
< gl,...,gn+ll
representing
rl,...,rn >,
the
I(S
3
K).
If G has the deflclency-I presentation
we will abuse notation and use gl and
We restrict
group elements.
corresponding
rj
to denote loops
consideration
to
X
presentations having representative loops gl and rj such that there are disks Dj
of
number
nKnlmum
s(K)
Let s
for all i,J.
and
wlth
gl
Dj
rj
int(Dj)
CI(N(
X
0
generators, Is a handlebody of genus s+l.
relators in such a presentation.
gl)),
a neighborhood of a "bouquet" of
t(K) be the tunnel number of K, which Is the minimum number of arcs that
need to be attached to K so that the complement is an open handlebody. Equivalently,
t(K) is the minimum number of l-handles (tunnels) that need to be removed from the
t(K) is also called the
knot exterior to obtain a Heegard decomposition of S 3.
Let t
"Heegard" genus of K [3,2].
J. ERBLAND AND M. GUTERRIEZ
290
2.
THEOREM: s(K)
L(K) for any tame knot K.
The fundamental group of the complement of K
t(K) is easy:
The direction s(K)
For each arc, a loop
t(K) arcs is free on t+l generators.
around that arc provides a relation. Such a presentation Is called geometric.
together with the t
Outline of proof, of t(K) g sg): For each
3
the corresponding disk
rj,
Dj
can be
We build up a space X s from X0 by stages
3Dj _3X0,
0.
using 2-handle surgeries along these disks; Xj is the result of the jth surgery. Xs
X
N’(K), a solid torus neighborhood of
fs the result of the last surgery, with S 3
moved so that
K.
int
(Dj)_S
X
s
Thus a Heegard decomposition of S 3 can be obtained by drilling tunnels through the
dlsks used in the 2-handle surgeries.
The rest of this paper is concerned with filling in the details.
3.
THE CONSTRUCTION.
XoUCI(N(DI))
Let X
by 2-handle surgery along
<
DI;
D
I’
D I.
For each
J > I,
(Fig.
then modify Dj to remove intersections.
I
J, if N(D i
Of course, there may be more intersections, but they must all be simple arcs
for each
la,b.
Dj
In the latter case, the Intersections can simply be "capped off".
or circles.
In the
case of multiple arcs, start with the outermost arc of the new disk and continue until
the
of
all
Intersections
have
removed.
been
increase in the number of disks.
The modification may result
consider a disk to be trivial If the union of the disk with
contalns an Incompressible 2-sphere.
in
out".
If any disk is "trivial", simply "throw it
an
We
and previous disks
Xj_
The determination of which disks are trivial is
has been modified, If extra dlsks result, then select one at a
If you get an Incompressible sphere, discard that disk from consideration and
after
not unlque:
tlme.
Dj
move on to the next.
Let
trivial
be the union of disks obtained from
Dj
>
I, let Xj be the space obtained from Xj_ by 2-handle
By mlnlmality of s, at least s 2-handle surgeries are
For j
disks.
surgeries
along
D’j.
The boundary of
performed.
after modification then removal of
Dj
Xj
differs from that of
elther,by a decrease in the
Xj_
genus of one component or by an increase in the number of components.
time none of the boundary components can be a 2-sphere.
Xs
construction
is bounded by torl only:
X
T
T
O
k.
We need to show that
k
0, where T O bounds a solid torus N’(K).
4.
Before finishing the proof, we make some essential observations.
(1)
7
I(xs)
7!(X)
At the same
Thus the end result of the
G, the isomorphism being induced by inclusion,
follows
from a standard Selfert-Van Kampen argument.
(ll) There is no 2-sphere
SXs
that separates two of these torl.
Thls follows
from the construction itself.
(ill)For
each
I,
l(Ti)/ l(Xs)
is
inJective.
Otherwise
.
there
exists
an
D
in T I and a disk D in X s bounded by
{-I,I} together with
and separates it
from
the
moved
torus
away
be
can
is
a
the
that
of
sphere
torus
part
essential loop
GEOMETRIC PRESENTATIONS OF CLASSICAL KNOT GROUPS
291
from the other tort, contradicting remark (li).
3
(iv) Each torus bounds a solid torus in S on one side and hence a (possibly
If
trivial) knot exterior on the other [4].
> O, then the solid side of T must
contain the knot;
(v)
I, let
For i
there exists a contractible loop in the exterior of K
otherwise,
Xs,
which is not contractible in
BI=
thus contradicting (i).
S3-Xs
of
component
bounded
by
T i.
B0
Let
be
the
component of
X
X
hence
5.
s
is
inJective.
O, we
l(Ti)/ l(Xs)
l(Xs)
l(Xs BI).
hypothesis
of
application
the
apply
and
I(Wj)"
t(Xs
is
for i
>
>
I(To) I(Bo)
is
inJective.
0.
Kampen
Selfert-Van
for
exterior
In particular, Wk
O.
Theorem
to get injectlvity of
X.
to
the
injectivlty
I(BI)/ l(Xs
B i) and
InJective.
l(Xs) I(Wo)
is
together with another
inJectlve,
I(Wj_I)
Kampen, to get InJectlvlty of I (Wj_I)
I (Wj) and
In particular,
that
knot
a
For related reasons,
I(TI)/ l(Bi)
We apply the
is
l(Xs)
Selfert-Van
"l(Xs U Bj)
Bi
(iv)
By
We will now proceed to show that B I
Let W
X
Bj for j
s B0, Wj --Wj_
0
inductive
of
N(K).
O
l(Ti)/ l(Bi)
For i
of
by T
bounded
Since Wk
X, we have
l(x)
l(Wo)* l(wl
G.
Since this composition is an isomorphism, all of the component injections must also be
Isomorphisms.
l(Xs)
From this and the Selfert-Van Kampen construction, it follows that
I(Xs
knot exterior.
B i) and hence
l(Bi)
for i
)
O, which is impossible for a
Thus only B 0 is non-empty.
It remains to verify that tnt(B 0
thus completing the proof.
6.
The
l(Ti)
V)
boundary of B 0 is the union of
N’(K), Just
two
a fatter neighborhood of K,
tort, one of which ts BN(K).
inclusions in both cases must Induce isomorphisms of fundamental groups.
I(Bo)
I(T0)
I(BN(K))
Z + Z is
abelian,
the
Hurewlcz
The
Since
homomorphlsm
is
an
292
J. ERBLAND AND M. GUTERRIEZ
isomorphism onto
HI(B0).
It follows that the merldional loops m and m 2 on the
Let a be a path from the base point of m to the base
bounding tori are homologous.
Then
point of m 2.
mlm2 a is homogous to 0 and hence homotoplcally trivial. Thus
the meridians bound an annulus and the torus T O is paraltel to N(K), which suffices
to complete the proof.
REFERENCES
I. MONTESINOS,
J.M.
Note
on
a
result
of
Boileau-Zieschang,
in
Low-dimensional
Topology and Klelnlan Groups, LMS Lecture Notes Set. 112 (1986), 241-252.
On Heegard Decompositions of Torus Knot
2. BOILEAU, M., ROST, M., ZIESCHANG, H.
Exteriors and Related Selfert Fibre Spaces, Math. Ann. 279 (1988), 553-581.
The Heegard genus of manifolds obtained by surgery on links and knots
Int. J. Math. & Math. Scl. 3 (1980), 583-589.
3. CLARK, B.
4. ROLFSEN, D.
Knots and Links, Publish or Perish Inc. (1976).