Complementary legs and rational balls

0.22 MB·

by  Ana G. Lecuona

COMPLEMENTARY LEGS AND RATIONAL BALLS
ANAG.LECUONA
7
1
0
2

n Abstract. InthisnotewestudytheSeifertrational homologyspheres with
a two complementary legs, i.e. with a pair of invariants whose fractions add
J
up to one. We give a complete classification of the Seifert manifolds with 3

8 exceptionalfibersandtwocomplementarylegswhichboundrationalhomology
balls. TheresulttranslatesinastatementontheslicenessofsomeMontesinos

] knots.
T
G
. 1. introduction
h
t Seifert manifolds are a particularly well understood set of 3–manifolds. They
a
m are oriented, closed 3–manifolds admitting a fixed point free action of S1 and they
are classified by their Seifert invariants [14]. They include all lens spaces. They
[
were first introduced in [16] and constitute one of the building blocks in the JSJ

1 decomposition. Theyhavebeenstudiedfromthemostvariousperspectivesandare
v
partiallyorcompletelyclassifiedfrommanydifferentpointsofview: Steinfillability,
0
symplectic fillability, tightness etc.
8
9 A relevant and difficult question one can ask about a 3–manifold is whether or
1 notitboundsarationalhomologyball[7]. InthecaseofSeifertmanifoldsthereare
0 somepartialanswersspreadoutintheliterature[3,6,9,2]andthereis acomplete
1. classification for the lens space case [10].
0 Seifert manifolds can be realized as the double cover of S3 branched over Mon-
7 tesinos links [12]. If a Seifert manifold does not bound a rational homology ball,
1
thenitisawellknownfact[5,Lemma17.2]thatthecorrespondingMontesinoslink
:
v does not bound an embedded surface in B4 with Euler characteristic equal to one.
i In the case of the link being a knot, this tells us that the knot is not slice. The
X
famous slice–ribbon conjecture states that every slice knot bounds an immersed
r
a disk in S3 with only ribbon singularities. A surface singularity is ribbon if it is the
identification of two segments, one contained in the interior of the surface and the
other joining two different boundary points.
Among the Seifert fibered rational homology spheres Y with Seifert invariants
Y = Y(b;(α ,β ),...,(α ,β )), b ∈ Z, α > β > 0, we will be interested in this
1 1 n n i i
note in those having a pair of complementary legs, that is, a pair of invariants
verifying βi + βj =1. The case n≤3 is of particularinterestsince for n≤3 there
αi αj
is a one to one correspondence between the set of Seifert manifolds and the set of
the correspondingMontesinoslinks andour results will haveaninterpretationalso
in terms of those. Our main result is the following.
Theorem1.1. The Seifert fiberedmanifold Y =Y(b;(α ,β ),(α ,β ),(α ,β ))with
1 1 2 2 3 3
β2 + β3 =1 is the boundary of a rational homology ball if and only if Y belongs to
α2 α3
the list L.
1
2 ANAG.LECUONA
ThelistL inthetheoremiscompletelyexplicitandcanbefoundinSection3.1.
Givenany setof Seifertspaces with atmost3 exceptionalfibers we will simply say
that a Montesinos link belongs to this set if the double cover of S3 branched over
this link belongs to it. In terms of Montesinos links we have the following result.
Theorem1.2. EachMontesinoslinkinthesetL istheboundaryofaribbonsurface
Σ such that χ(Σ)=1.
An immediate corollary of this theorem is that the slice–ribbon conjecture is
true inside the family of Montesinos knots whose double covers are Seifert fibered
spaces with 3 exceptional fibers and two complementary legs.
The proof of Theorem 1.1 relies very heavily on the analysis previously done in
[10, 8]. In this note we first show that a Seifert space Y with complementary legs
is rationalhomology cobordantto a lens space L. This allows us to study a lattice
embedding problem in a veryefficient way,which yields a relationshipbetween the
SeifertinvariantsofY andofL. The classificationoflens spacesbounding rational
homology balls in [10] allows us to make the list L explicit.
1.1. Organization of the paper. In Section 2 we introduce the strict minimum re-
gardingSeifertmanifolds,thelanguageoflatticeembeddingsandMontesinoslinks.
InSection3we carryoutthe proofsofthe theoremsinthe introductionandestab-
lish the list L. Finally in Seciton 4 we collect some noteworthy technical remarks
about the lattice embedding problem.
2. Notation and Conventions
2.1. Graphs and Seifert spaces. It is well known [13] that, at least with one of the
twoorientations,aSeifertfiberedrationalhomologyspheredefinedbytheinvariants
Y = Y(b;(α ,β ),(α ,β ),(α ,β )) is the boundary of a 4–manifold obtained by
1 1 2 2 3 3
plumbing according to the graph:
−a −a
1,2 n2,2
• ... •
−an1,1 −a1,1 −a0
• ... • •
−a −a
1,3 n3,3
• ... •
where
α 1
i
=[a ,...,a ]:=a − , a ≥2 and a =−b≥1.
βi 1,i ni,i 1,i 1 j,i 0
a −
2,i
...
1
a −
ni−1,i a
ni,i
This graph is unique and will be called the canonical graph associated to Y. We
will use the expression three legged canonical graph to denote any such graph.
If we are given a weighed graph P, like for example the one above, we will
denote by M the 4–manifold obtained by plumbing according to P and by Y
P P
its boundary. The incidence matrix of the graph,which represents the intersection
pairing on H (M ;Z) with respect to the natural basis, will be denoted by Q .
2 P P
COMPLEMENTARY LEGS AND RATIONAL BALLS 3
The number of vertices in P coincides with b2(MP) and we will call (Zb2(MP),QP)
the intersection lattice associated to P. The vertices of P, which from now on will
beidentifiedwiththeirimagesin Zn andwillalsobe calledvectors,areindexedby
elements of the set J :={(s,α)|s∈{0,1,...,n }, α∈{1,2,3}}. Here, α labels the
α
legsofthegraphandL :={v ∈P|i=1,2,...,n }isthesetofverticesoftheα-
α i,α α
leg. The string associatedto the leg L is the n -tuple of integers (a ,...,a ),
α α 1,α nα,α
where a := −Q (v ,v ) ≥ 0. The three legs are connected to a common
i,α P i,α i,α
centralvertex,whichwe denote indistinctly by v =v =v =v (notice that,
0 0,1 0,2 0,3
with our notation, v does not belong to any leg).
0
Let (Zn,−Id) be the standard negative diagonal lattice with the n elements of
a fixed basis E labeled as {e }n . As an abbreviation in notation we will write
j j=1
ej · ei to denote −Id(ej,ei). If the intersection lattice (Zb2(MP),QP) admits an
embedding ι into (Zn,−Id), we will then write P ⊆ Zn and will omit the ι in the
notation,i.e.instead ofwriting ι(v)= x e we will directly writev = ,x e .
Pj j j Pj j j
We will adopt the following notation: for each i ∈ {1,...,n}, (s,α) ∈ J and
S ⊆P we define
V :={j ∈{1,...,n}| e ·v 6=0 for some v ∈S}.
S j s,α s,α
Givene∈Zn withe·e=−1,wedenote byπ :Zn →Zn the orthogonalprojector
e
onto the subspace orthogonalto e, i.e.
π (v):=v+(v·e)e∈Zn, ∀v ∈Zn.
e
2.2. Complementary legs. The key concept in this paper is that of complemen-
tary legs. As explained in the introduction, they are pairs of legs in the graph P
associated to Seifert invariants with the property βi + βj =1.
αi αj
Suppose that in the graph displayed at the beginning of Section 2.1 the legs
L and L were complementary. It is well known then that the strings of integers
2 3
(a ,...,a ) and (a ,...,a ) are relatedto one another by Riemenschneider’s
1,1 n1,1 1,2 n2,2
point rule [15] and that there is an embedding of L ∪L ⊆ Zn2+n3 [11]. There
2 3
essential features of this embedding that we will need in this note are:
• There exists a sequence ofcontractions(see [10] for the definition andgen-
eraltheory ofembeddings oflinear sets) to the legsL˜ andL˜ with associ-
2 3
ated strings (2) and(2). The embedding L˜ ∪L˜ ⊆Z2 is unique andgiven
2 3
by e +e and e −e .
1 2 1 2
• Thebasisvectore appearsonlytwiceintheembeddingL ∪L ⊆Zn2+n3.
1 2 3
• Given a linear weighed graph L with string (b ,...,b ) and a embedding
1 k
L⊆Zk,itispossibletocombineLandtwocomplementarylegstoyielda3
leggedgraphP withaembeddinginZk+n2+n3. Indeed,thethreelegswill
L
bethecomplementarylegsL andL ,andthelegL withassociatedstring
2 3 1
(b ,...,b ); thecentralvertexwillhaveweightb +1andtheembedding
1 k−1 k
is the obvious one, adding −e to the embedding of the vertex in L with
1
weight b .
k
• The embedding in Zk+n2+n3 of the graph P constructed in the preceding
L
point can be contracted to an embedding of Pc ⊆ Zk+2 which is a graph
L
with L˜ and L˜ as complementary legs.
2 3
A final notion that will appear in the text and is related to complementary legs
isthatofthedual ofastringofintegers(a ,...,a ),a ≥2. Itreferstotheunique
1 n i
4 ANAG.LECUONA
string (b ,...,b ), b ≥2 such that
1 m i
1 1
+ =1.
[a ,...,a ] [b ,...,b ]
1 n 1 m
2.3. Montesinos links. Seifert spaces admit an involution which presets them as
double covers of S3 with branch set a link. This link can be easily recovered
from any plumbing graph P which provides a surgery presentation for the Seifert
manifoldY . AsshowninFigure1theKirbydiagramobtainedfromP isastrongly
P
invertiblelink. TheinvolutiononY ,whichisarestrictionofaninvolutiononM ,
P P
yieldsasquotientS3,boundaryofB4,andthebranchsetconsistsofacollectionof
twistedbandsplumbedtogether. Thetwistscorrespondtotheweightsinthegraph
and the plumbing to the edges. The Montesinos link is by definition the boundary
of the surface obtained by plumbing bands [12]. An example is shown in Figure 1.
Givena threeleggedgraphP,thereisaunique Montesinoslink ML whichcan
P
be obtained by the above procedure. If we start with a linear graph P′, then the
same procedure yields a 2-bridge link, which we will denote by K(p,q) where YP′
is the lens space L(p,q).
3. The list L and consequences
The key feature of plumbing graphs with complementary legs is the following
proposition. A similar proof in a more generalsetting can be found in [1, Proposi-
tion 4.6] (cf. [9, Remark 6.5]).
Proposition 3.1. Let P be a 3 legged canonical graph with two complementary
legs L and L and such that the string associated to L ∪ {v } is of the form
2 3 1 0
(a ,...,a ,a ). Then, there is a rational homology cobordism between Y and
n1,1 1,1 0 P
the 3–manifold YP′ associated to the linear graph with string (an1,1,...,a1,1,a0−1).
Proof. Since the graph P has two complementary legs, one can attach a 4 dimen-
sional2−handlehtoMP obtaininga4−manifoldwhoseboundaryis(S1×S2)#YP′,
where P′ is a linear graph with associated string (a ,a ,...,a ,a −1) [8,
n1,1 n1−1,1 1,1 0
Lemma3.1]. Noticethatthedual2−handleh˜,attachedto(S1×S2)#YP′,killsthe
freepartofH1((S1×S2)#YP′;Z), since YP =∂MP isa rationalhomologysphere.
Moreover, attaching a 3−handle k to (S1 ×S2)#YP′ along one of the essentials
spheres of the summand S1×S2, the free part of H2((S1×S2)#YP′;Z) is killed
andone obtains YP′. Dually, attaching a 1−handle k˜ to YP′ yields (S1×S2)#YP′.
Itfollowsthat,ifYP′ boundsarationalhomologyballW,thenYP isthebound-
ary of the 4−manifold X := W ∪ k˜ ∪ ˜h, where the attaching sphere of h˜ has
non-trivial algebraic intersection with the belt sphere of k˜, since h˜ kills the free
part of H ((S1 ×S2)#Y ;Z). We claim that X is a rational homology 4−ball.
1 S
In fact, since W is a rational homology ball, we have H (X;Q) = H (X;Q) = 0.
4 3
The rest of the homology groups can be easily computed using cellular homology
with rational coefficients: calling Xm the m−skeleton of the CW–complex X, the
cellular chain complex is
··· //H (X3,X2;Q) d3 //H (X2,X1;Q) d2 // H (X1,X0;Q) d1 //Q .
3 2 1
Since the attaching sphere of h˜ ∈ H (X2,X1;Q) has non-trivial algebraic inter-
2
section with the belt sphere of k˜ ∈ H (X1,X0;Q), by definition of the boundary
1
operatord , we haveh˜ 6∈Kerd , andsince W is a rationalhomologyball it follows
2 2
COMPLEMENTARY LEGS AND RATIONAL BALLS 5
−1
2 −2
−1
−1 −1 u 3 −2
2
3
(a)
=∼
(b)
Figure 1. Part (a) shows a 3 legged plumbing graph and its associated Kirby diagram
as a strongly invertible link. The bottom picture shows the branch surface in B4 which
is the image of Fix(u). This surface is homeomorphic to the result of plumbing bands
according to theinitial graph: thegray lines retract onto thesides of therectangle.
that H (X;Q) = 0. Finally, we assume, without loss of generality, that X is con-
2
nected and hence d ≡ 0. Therefore, we have H (X;Q) = H (X1,X0;Q)/imd .
1 1 1 2
ThegroupH (X1,X0;Q),whichisadirectsumoffinitelymanyQ,hasonemoreQ
1
summand, comingfromthe 1−handle k˜, than H (W1,W0;Q). However,k˜∈imd
1 2
andthereforeH (X;Q)=H (W;Q)=0. Infact, anappropriatedecompositionof
1 1
X and W gives the following commutative diagram.
H (W2,W1;Q) d2 // H (W1,W0;Q) d1 // 0
2 1
i∗ i∗
(cid:15)(cid:15) (cid:15)(cid:15)
H (X2,X1;Q) d2 //H (X1,X0;Q)
2 2
Note that the first row of the diagram is exact because W is connected and
H (W;Q) = 0. Since the attaching sphere of h˜ has non-trivial algebraic inter-
1
section with the belt sphere of k˜, there exists some α ∈ H (W1,W0;Q) such that
1
d (h˜)=qk˜+i (α), where q ∈Q, q 6=0. The exactness of the firstrowimplies that
2 ∗
6 ANAG.LECUONA
α=d (β) for some β ∈H (W2,W1;Q). Therefore
2 2
d (h˜−i (β))=d (h˜)−d i (β)=qk˜+i (α)−i d (β)=qk˜,
2 ∗ 2 2 ∗ ∗ ∗ 2
and, as claimed, X is a rational homology ball.
Vice versa, if YP bounds a rational homology ball Z, then YP′ is the boundary
of a 4−manifold V := Z ∪h∪k, where this time the attaching sphere of k has
non-trivial algebraic intersection with the belt sphere of h. It follows, arguing as
before,thatV isarationalhomologyball. Weconcludethatthereexistsarational
homology cobordism between YP and YP′. (cid:3)
IfP isacanonical3leggedgraphandY istheboundaryofarationalhomology
P
ball W, then there is an embedding of the lattice associated to H (M ,Z) into
2 P
the standard negative lattice of the same rank. Indeed, since P is canonical the
intersectionformQ ofM isnegativedefinite[13,Theorem5.2]andhence,X :=
P P P
M ∪ (−W) is a closed smooth negative 4-manifold. By Donaldson’s Theorem
P YP
[4] the intersection lattice of X is isomorphic to (Zn,−Id), where n = b (X ).
P 2 P
Moreover, since W is a rational homology ball n = b (M ) and therefore, via X,
2 P
there is an embedding of (H (M ,Z),Q ) into (Zn,−Id), which we simply write
2 P P
asP ⊆ZP. Inthenextlemmawestudytheconsequencesofthisobstructionwhen
the graph P has two complementary legs.
Lemma 3.2. Let P be a canonical 3 legged graph with two complementary legs L
2
and L and suppose that Y is the boundary of a rational homology ball. Then, the
3 P
linear set L ∪{v } has associated string of length n +1 of the form
1 0 1
(a ,...,a ,2[r]),
n1,1 r,1
where 0≤r ≤n +1 and the string(a ,...,a −1)corresponds toa linear graph
1 n1,1 r,1
admitting an embedding in Zn1−r+1 and defining a lens space LP′.
Proof. Since YP bounds a rational homology ball, the space YP′ from Proposi-
tion 3.1 does too and therefore it holds P′ ⊆ZP′. From Section 2.2 we know that
itisalwayspossibletoextendtheembeddingofP′ ⊆ZP′ toanembeddingP ⊆ZP
insuchawaythatV ∩V =∅. ItfollowsthatifY boundsarationalhomol-
L1 L2∪L3 P
ogy ball then there is an embedding P ⊆ZP with the property V ∩V =∅
L1 L2∪L3
which allows us to contract the complementary legs to L = {v = e +e } and
2 1,2 j k
e
L ={v =e −e } for some j,k ∈{1,...,n}.
3 1,3 j k
e
Since v ·v = v ·v = 1, then j ∈ V , |e ·v | = 1 and k 6∈ V . If v ∈ L
0 1,2 0 1,3 v0 j 0 v0 1
wassuchthatV ∩{e ,e }6=∅,thenwecouldnothavev·v =0andv·v =0,
v j k 1,2 1,3
therefore V ∩V =∅. Since v ·v =1, we have V ∩V 6=∅ and hence
L1 L2∪L3 1,1 0 v0 L1
v =v˜ −e with V ⊆V . Moreover,it follows that a =−v ·v ≥2.
0 0 j v˜0 L1 0 0 0
Ifa ≥3thenv˜ ·v˜ ≤−2. Bydefinitonofcomplementarylegs|V |=n +n
0 0 0 L2∪L3 2 3
and therefore P′ := L ∪{v˜ } ⊆ Zn1+1. Notice that in this case the r in the
n1+1 1 0
statement equals 0.
If a = 2 let us call 1 ≤ r ≤ n +1 the bigest integer such that a = ... =
0 1 0,1
a = 2. It is not difficult to check that the r vectors v ,...,v belong to
r−1,1 0,1 r−1,1
thespanofr+1basisvectors. Ifr =n +1,then|V |=n +2,bydefinition
1 L1∪{v0} 1
of complementary legs |V |=n +n and V ∩V ={j}. Therefore
L2∪L3 2 3 L1∪{v0} L2∪L3
every 3 legged canonical graph with two complementary legs L and L satisfying
2 3
a =2 ∀j ∈{0,...,n }admits anembedding intoZn,where n=n +n +n +1.
j,1 1 1 2 3
COMPLEMENTARY LEGS AND RATIONAL BALLS 7
Finally,ifr <n +1thereexistsi∈{1,...,n}suchthat{i}=V ∩V and
1 vr,1 vr−1,1
since |V ∩V |=1 we have that
L1 v0
r+1
Pn′1−r+1 :=(L1\ [ vj−1,1)∪πei(vr,1)⊆Zn1−r+1.
j=2
Since v ·v = 1, we have π (v )·π (v ) = −a +1 and the statement
r−1,1 r,1 ei r,1 ei r,1 r,1
follows. (cid:3)
Combining Proposition 3.1 and Lemma 3.2 and using the same notation and
conventions we immediately obtain:
Corollary 3.3. The 3–manifold Y bounds a rational homology ball if and only if
P
either r =n1+1 or the lens space LP′ bounds a rational homology ball.
Proof. From the proof of Lemma 3.2 we know that if r = n +1 then the string
1
associated to P′ is (2,2...,2,1) and therefore YP′ = S3 which obviously bounds a
rational homology ball. On the other hand, if r < n +1 then the string is of the
1
form (an1,1,...,ar,1,2,...,2,1) and therefore YP′ = LP′ where LP′ is a lens space
with associated string (a ,...,a −1). We conclude using Proposition 3.1. (cid:3)
n1,1 r,1
3.1. The List. The above analysis yields a complete list L of Seifert spaces with
3 exceptionalfibers andtwo complementarylegs bounding rationalhomologyballs
and proves Theorem 1.1. Every element in L is, with one of the two orientations,
the boundary of a 4–manifold obtained by plumbing according to a graph
−a −a
1,2 n2,2
• ... •
−a −a −2 −2
n1,1 r,1
• ... • • ... •
−a −a
1,3 n3,3
• ... •
where the strings (a ,...,a ) and (a ,...,a ) are related to one another by
1,2 n2,2 1,3 n3,3
Riemenschneider’spointrule,thenumberof−2intheleftlegisarbitrary(including
the possibility zero) and the string (a −1,...,a ), up to order reversal and
r,1 n1,1
duality, can be found in [8, Remark 3.2] which summarizes [10, Lemmas 7.1, 7.2
and 7.3].
3.2. Montesinos links and ribbon surfaces. Given that there is a one-to-one corre-
spondence between the set of Seifert spaces with at most 3 exceptional fibers and
the Montesinos links which arise as branch sets, we can translate the analysis on
the rational homology balls developed so far into the language of Montesinos links
andribbonsurfaces. Theorem1.2inthe introductionis animmediate consequence
of the following proposition.
Proposition 3.4. Let P be a canonical graph with two complementary legs. The
Montesinos link ML ⊂S3 is the boundary of a ribbon surface Σ with χ(Σ)=1 if
P
and only if the Seifert space Y is the boundary of a rational homology ball.
P
Proof. Since P has two complementary legs, we know that there is a ribbon move
that changes the link ML into the disjoint union K(p,q)⊔U, where U stands for
P
8 ANAG.LECUONA
the unknot and K(p,q) is the 2-bridge link associated to the lens space LP′ in the
statement of Lemma 3.2 [8, Lemma 3.3 and discussion before it].
If YP is the boundary of a rationalhomologyball, then, by Proposition3.1, LP′
is also the boundary of a rational homology ball. [10, Theorem 1.2] tells us then
that K(p,q) ⊂ S3 is the boundary of a ribbon surface Σ′ with χ(Σ′) = 1. The
statement follows immediately.
Conversely, if there exist a surface Σ and a ribbon immersion Σ # S3, with
∂Σ = ML and χ(Σ) = 1, then it is well known that Y is the boundary of a
P P
rational homology ball [5]. (cid:3)
In this last proposition we have shown that any 3 legged slice Montesinos knot
with two complementary legs is actually the boundary of a ribbon disk. We have
thus:
Corollary 3.5. The slice-ribbon conjecture holds true for all the Montesinos knots
ML with P a 3 legged canonical graph with 2 complementary legs.
P
4. noteworthy remarks on lattice embeddings
• Similar families of 3 legged graphs have been studied in [8]. In that article
all graphs Γ such that Γ ⊆ ZΓ define Seifert spaces which bound rational
homology balls and the corresponding Montesinos links are the boundary
of ribbon surfaces with Euler characteristic 1. In other words, Donaldson
obstruction is sufficient to yield a complete classification. The situation
is completely analogous in the case of linear graphs (lens spaces) [10]. In
contrast, not all the sets studied in this article, that is 3 legged graphs P
withtwocomplementarylegssuchthatP ⊆ZP,defineSeifertspaceswhich
bound rational homology balls. For example, the Seifert space associated
to
e1−e2
e5−e4 −e5+3e6 −e3+e4+e5
−e1+e3
e1+e2
is not the boundary of a rational homology ball, since the lens space
L(36,19) with associated string (2,10,2) does not belong to [10, Theo-
rem 1.2].
• In all the cases studied in [8, 10, 6, 9] if a graph Γ yields a 3–manifold Y
Γ
which bounds a rational homology ball, then all the coefficients appearing
in the embedding Γ ⊆ ZΓ can be chosen to be equal to ±1. This is no
longer true in the family studied in this note. For example,
e1−e2
2e4−e3
e3−e1
e1+e2
satisfies |v ·e | = 2 and it is the boundary of a rational homology ball,
1,1 4
since the lens space L(4,1) has this property.
COMPLEMENTARY LEGS AND RATIONAL BALLS 9
References
1. Paolo Aceto, Rational homology cobordisms of plumbed 3 manifolds, arXiv preprint:
1502.03863.
2. Paolo Aceto and Marco Golla, Dehn surgeries and rational homology balls, arXiv preprint:
1509.07559.
3. Andrew J. Casson and John L. Harer, Some homology lens spaces which bound rational ho-
mology balls,PacificJ.Math.96(1981), no.1,23–36. MR634760
4. S. K. Donaldson, The orientation of Yang-Mills moduli spaces and 4-manifold topology, J.
Differential Geom.26(1987), no.3,397–428. MR910015
5. D.EisenbudandW.Neumann,Three-dimensional link theory and invariants of plane curve
singularities,AnnalsofMathematicsStudies,vol.110,PrincetonUniversityPress,Princeton,
NJ,1985. MR817982(87g:57007)
6. JoshuaGreeneandStanislavJabuka,Theslice-ribbonconjecturefor3-stranded pretzelknots,
Amer.J.Math.133(2011), no.3,555–580. MR2808326
7. RobKirby, Problems in low dimensional manifold theory, Algebraicandgeometric topology
(Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos.
PureMath.,XXXII,Amer.Math.Soc.,Providence, R.I.,1978,pp.273–312. MR520548
8. Ana G. Lecuona, On the slice-ribbon conjecture for Montesinos knots, Trans. Amer. Math.
Soc.364(2012), no.1,233–285. MR2833583
9. , On the slice-ribbon conjecture for pretzel knots, Algebr. Geom. Topol. 15 (2015),
no.4,2133–2173. MR3402337
10. Paolo Lisca, Lens spaces, rational balls and the ribbon conjecture, Geom. Topol. 11 (2007),
429–472. MR2302495
11. , Sums of lens spaces bounding rational balls, Algebr. Geom. Topol. 7 (2007), 2141–
2164. MR2366190
12. JoséM.Montesinos,Seifertmanifoldsthatareramifiedtwo-sheetedcycliccoverings,Bol.Soc.
Mat.Mexicana(2)18(1973), 1–32. MR0341467
13. Walter D.NeumannandFrankRaymond, Seifert manifolds, plumbing, µ-invariant and ori-
entationreversingmaps,Algebraicandgeometrictopology(Proc.Sympos.,Univ.California,
SantaBarbara,Calif.,1977),LectureNotesinMath.,vol.664,Springer,Berlin,1978,pp.163–
196. MR518415
14. Peter Orlik and Frank Raymond, On 3-manifolds with local SO(2) action, Quart. J. Math.
OxfordSer.(2)20(1969), 143–160. MR0266214
15. Oswald Riemenschneider, Deformationen von Quotientensingularitäten (nach zyklischen
Gruppen), Math.Ann.209(1974), 211–248. MR0367276
16. H. Seifert, Topologie Dreidimensionaler Gefaserter Räume, Acta Math. 60 (1933), no. 1,
147–238. MR1555366
Institute of Mathematics,Universityof Aix-Marseille,France
E-mail address: [email protected]