Intern. J. Math. & Mh Sci.
Vol. 3 No. 3 (1980) 583-589
583
THE HEEGAARD GENUS OF MANIFOLDS OBTAINED BY SURGERY
ON LINKS ANO KNOTS
BRADD CLARK
Department of Mathematics
University of Southwestern Louisiana
Lafayette, Louisiana 70504
U.S.A.
(Received January 15, 1979)
ABSTRACT.
Let L c S 3 be a fixed link.
It is sbuwn that there exists an upper
bound on the Heegaard genus of any manifold obtained by surgery on L.
number of L,
T(L),
of the knot K,
If K’
is defined and used as an upper bound.
it is shuwn that T(K’) < T(K) + I.
If M
The tunnel
is a double
is a manifold
obtained by surgery on a cable link about K which has n ccmponents, it is
shown that the Heegaard genus of M is at most T(K) + n + i.
1980 ICS SLJECT CLASS IF ICATI( CODES.
KEY 3RDS AND PHRASES.
Lickorish in
[5
Primary 57M25, 57NI0.
Links, Krts, Heegaard genus, 3-manifold.
demonstrated that all ccapact Oonnected orientable
3-manifolds without boundary could be represented as a result of surgery on a link
in S 3.
Since then, considerable wrk has been done on the so-called closed
3-manifolds by studying various knots and links in
S 3.
B. CLARK
584
Both in
[3]
and
[8]
a relationship between a knot and the Heegaard splitting
of manifolds obtained by surgery on that knot was hinted at.
6]
that the Mazur hcology sphere (see
[i] to
A similar process w-as used in
[ 2 ],
In
Dibner
has a genus-2 Heegaard splitting.
show that n + 1
is an upper bound to the
genus of a manifold obtained by surgery on a torus link with n ccmponents.
The purpose of this paper is to extex these tec/miques to other knots and links
and thereby obtain a bound on the Heegaard genus of manifolds obtained by surgery
on those links and knots.
If X is a point set, we shall use
for the closure of X,
and
int(X)
cl(X)
for the interior of X,
X for the boundary of X.
The genus of a manifold
is defined to be the minimal genus of a Heegaard splitting of the manifold.
N(X) c M is called a
X be a polyhedron oontained in the P.L. n-manifold M.
regular neighborhood of X in M if X oN(X)
Let
is an n-manifold
and N (X)
which can be simplicially ollapsed to X.
As such, all manifolds
This paper deals with piecewise linear topology.
are onsidered to be simplicial and maps to be piecewise linear.
ass
This will be
as additional hypotheses throughout the paper.
Let L be a link in
o projection.
P
3.
Let P be any plane and p
neither is a vertex of L.
/
if for every x e P,
is regular for L
onsists of at most tw points; and if
R3
p-l(x)
p
the
-I (x)
L
p
L does ontain tw points,
The crossing number for L,
c(L),
is the min
number of double-points in any regular projection of any link L’
of the same
is splittable if there exist disjoint 3-cells B I and
2
B2 with LI tint B I and L2 cint B 2. Thus if L is any link we can write
A link L
L
L
I
L
L where Lic int
n
unsplittable for i < i < n.
L1
Bi,
Bi
Bj
@
for
i
j
and Li
is
585
HEEGAARD GENUS OF MANIFOLDS ON LINKS AND KNOTS
Let L be a link in S3
cl(S 3
We associate with L the manifold C
DEFINITION.
t cC
a regular neighborhood of L in S 3.
and N(L)
t
is a tunnel if
N(L)).
C is a pair of
t
is a 3-cell and
disjoint 2-eells.
THEOREM 2.1.
L
Ln,
L
1
PNOOF:
If M is a 3-manifold obtained by surgery on
n
then the genus of M is at most
c(Li) + n.
M can be written as M1 #
obtained by surgery on the link Li.
i=17
#
By 4 we knc that the genus of a
onnected sum is the sum of the genera of the cponents.
to
that the genus of
If c(Li)
double points.
m,
is the manifold
where
nus
it will suffice
c(Li) + i.
is at most
we can find a regular projection for Li which has m
i"’" ’m
We can find
where
is the straight line segment
j
between the points of Li which fozm the j-th double point of the projection.
Let N(Li)
manifold.
be a regular neighborhood of Li
For each
neighborhood of
m
%i
in S 3,
we can find a tunnel
C and such that ti
ti
tj
and C
such that
@
for
m
its associated
t.1 is a regulr
i
j.
t N(Li) is a handlebody. Clearly U j U Li is a oore for
j
j=l
this handlebody and this ore can be deformed into the plane of projection.
H=
j=l
Therefore H’
cl(S 3
H)
is also a handlebody.
the ore of H has one zponent.
a simple Euler characteristic
After
Since L.1
defor
is unsplittable,
this ore into the plane,
.
t
that the genus of
H
,
is m + i.
We note that if N’ (Li) is a regular neighborPod of Li in
then
m
U tj U N’ (Li) and H’ form a Heegaard splitting of
j=l
The above theorem gives a rather crude estimate of the max genus of a
manifold obtained by surgery on a link.
As we shall see, in specific situations
we will be able to give a better estimate.
We also see that if L is an
unsplittable link, we can always find a set of pairwise disjoint tunnels which
can oonvert the associated manifold to a lebody.
B. CLARK
586
DEFINITION.
T(L),
is an unsplittable link, the tunnel number of L,
L
If
number of disjoint tunnels needed to onvert the associated
is the
manifold into a handle/xx.
In
7]
clear from this description that any pretzel knot
D1 and D2
points.
cl(S 3
Also
PNOPOSITION 3.1.
H)
K can be embedded in the
K and D2
so that D1
for H
It is
in such a way that there are cutting
boundary of a cube-with-)-handles H
disks
as pretzel knots.
H.F. Trotter described a class of knots
K each onsist of t
is itself a cube-with-t%-handles.
If M is obtained by surgery on a pretzel knot, then M
has Heegaard genus at most 3.
We assume that the pretzel knot K c H is as described above.
PROOF.
can find an arc eic Di with
ei
K
Di
for
i
2.
or
i
We
Without loss
of generality we my assume that a regular neighbo of the cutting disk
Di
in H is of the form D
If N(K)
H’
N(K)
B
cl(H-
(K
B)
U
iX
ad that K
I
is a regular neighborhood of
(D1 X I)
(D2
I)
[(mI I)
(D2
I)]).
{0,i})
(eI
(e
2
0 (Di
K in H,
I)
ei
then obviously
Let
is a cube-with-three-handles.
We ote that
{0,1})
is a simple closed curve in
Let D be the disk in B bounded by this simple closed curve.
with the
for
cl(S 3
cutting disks of cl (S 3
H’).
I.
H)
B
Then
S
2.
D along
will form a system of cutting disks
Using the same reasoning as in Torem 2.1, we have found a
genus 3 Heegaard splitting for any manifold obtained by surgery on a pretzel knot.
PROPOSITION 3.2.
knot KI # K2,
PROOF.
S
then the genus of M is at most T(KI) + T(K
2)
We can find a 2-sphere
separates S 3
two points.
If M is a manifold obtained by surgery on the
If
+ 2.
S c S 3 with the follcing properties.
into t%D 3-cells B
c S
site
1
and B2.
(K1 # K2)
is an arc connecting these tD points,
S
consists of
587
HEEGAARD GENUS OF MANIFOLDS ON LINKS AND KNOTS
[B1
.
K2)] e
(K1 #
knot of type
K2)]
e
(K #
and [B
2
1
1
Without loss of generality we my assume that S
I
is a knot of type
regular neighborhood of S
in
S3
K
(S I)
and that
(K
1
# K2)
is a
is a
I.
8
Since K’
[e {0}] is a knot of type KI,
[(K1 # X2) 0 cl(B1 S I)]
1
it has tunnel number T(KI). We can find a Heegaard splitting of B 1 of genus
T(KI)
knot of type K
2
[(K1 # K2) 0 cl(B2
K
+ I. Likewise since
it has tunnel number
splitting of B2 of genus
T(K2)
by surgery on K’,
I
is a double of K and M is a manifold obtained
is a double of
the follcing properties.
+ 2.
K we can find a singular disk D with
A copy of K is contained in D.
D and the only singularity in D is a single double arc
U) be a regular neighborhood
Let N(K’
D’
cl(D- N(K’
knot of type
e))
K’
e
K’
T(K) + i.
is spanned by
with
in S 3.
8 c K’.
We te that
is a solid torus with a
This means that if we add T(K)
we can obtain a Heegaard splitting of genus
from N(D)
of
is a real disk and that N(D)
K for a core.
This will yield
I.
in S
T(KI) + T(K2) + 2.
then M has genus at most T(K)
Since K’
PROOF.
is a
+ i. We can connect these splittings by adding
a Heegaard splitting of M of genus
If K’
{i}]
and we can find a Heegaard
T()
the 3-cell which is a regular neighborhood of
PROPOSITION 3.3.
I)]
S
ttu%nels to N(D)
When we rerme N(D’)
T(K) + 2.
we will obtain a Heegaard splitting of M of genus
In addition to these specific knots, we can find a better estimate than
Theorem 2.1 gives for the Heegaard genus of manifolds obtained by surgery on
specific links.
T3REM 4.1.
If M is a 3-manifold obtained by surgery along a cable link
about K with n nents, then M has genus at most T(K)
PROOF.
+n+
i.
Let T be a solid torus in S 3 which has K for a core.
T into 2n annuli
{,BI,,B2,... ,,n}
each follcing an
(x,y)
We split
curve
588
B. CLARK
on
T.
C
cl(S 3
T I be a oollar attached to T with T
Let
(T
(T
such that
ontained in C
We my also assume that
1/2, I] where
We can think of M as being obtained by surgery on Ai
Ic T XI.
Ai
Let Di be a disk in Ai with the property that cl(Ai
Let e i and
is a disk.
(cl(Ai- mi)
ei
(Di
c M
be
M after surgery is performed on the cable link.
I
We
that
[0,1/2]) T
(Di
is a 3-cell and
Ni are disks, we have that H1
is a cbe-with-n+l-handles.
[1/2,1])
h(Aix
Finally, we let Ni
8i"
[i/2,1] in
DiM [0,1/2] c T
[0,1/2])
Di)
8 i be arcs in Ai with the property that
the image of A
Since
Let
Without loss of generality, we may assume the
I))).
existence of T (K) tunnels tl,..., (K)
T(K)
ti) is a cube-with-handles.
cl(Ci=l
ti
Cc int(B1 X {l})c T I.
Ai
{0}.
T
T
T(K) ti
ni=l
and
[(Di [0,1/2])
Ni
2T(K)
C is a set of
i=l
pairwise disjoint disks ontained in int(B1
we can extend ti to the 3-cell
H
T(K)
t
ti
{I}).
(di, 1
I)
If
ti
C
di, 1 di, 2
I).
(di, 2
Then
I
t is a cube-with-n+T(K)+l-handles.
i=l
We note that since C is hcmecorphic to C
H1
n
cl[(C
that H2
Since e. X I
1
contained in
added to H2.
I
ni=l (Bi
and
T
8. I
I,
T(K)
t5’
j=l
are disks on H
I))
we have that cl(Ai
Thus M- int
H H
(Bi I), we must have
i=l
is a cube-with-T(K)+l-handles
2 where
Di)
ei x
I
Ic T I
and
8i
I
are
is a handle
is also a cube-with-n+T(K)+l-handles.
Clark, B. Surgery on Links Containing a Cable Sublink, Proc. Amer. Math. So_.
(to appear).
Dibner, S. Heegaard Splittings for an Infinite Family of Closed Orientable
3-manifolds, Ph.D. Thesis, State University of New York at Binghamton,
1977.
Haken, W. Various Aspects of the Three-dimensional Poincare Problem,
Topology of Manifolds, ed. J.C. Cantrell and C.H. Edwards, (Markham,
HEEGAARD GENUS OF MANIFOLDS ON LINKS AND KNOTS
589
Chicago) (1969) 140-152.
4.
Haken, W.
5.
Lickorish, W.B.R. A Representation of Orientable Omnbinatorial 3-manifolds,
Ann. of Math. 76 (1962), 531-540.
6.
Mazur, B.C. A Note on Scme Contractible 4-manifolds, Ann. of Math. 73 (1961)
221-228.
7.
Trotter, H.F.
8.
Waldhausen, F.
Scme Results on Surfaces in 3-manifolds, Studies in Modern
Topology, ed. P.J. Hilton, M.A.A. Studies in Math. Vol. 5, 39-98.
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