Canad.Math.Bull.Vol.46(3),2003pp.356–364
Branched Covers of Tangles in Three-balls
Makiko Ishiwata,J ´ozef
H.Przytycki and Akira Yasuhara Abstract.We give an algorithm for a surgery description of a p -fold cyclic branched cover of B 3
branched along a tangle.We generalize constructions of Montesinos and Akbulut-Kirby.
Tangles were first studied by Conway [4].They were particularly useful for analyz-ing prime and hyperbolic knots.A branched cover of the three-ball branched along a tangle (succinctly a branched cover of a tangle)is an indispensable tool for under-standing tangles.Hence it is important to give practical presentations of branched covers of tangles.Recall that a p -fold cyclic branched cover of a link or tangle (ori-ented for p >2)is uniquely defined by an epimorphism of the fundamental group of the complement onto Z p which sends meridians to 1.A p -fold branched cover of an n -tangle is a three-manifold,the boundary of which is a connected surface of genus (n −1)(p −1).Such a manifold can be obtained from the genus (n −1)(p −1)han-dlebody by a surgery.We provide an algorithm for a surgery d
escription of a p -fold cyclic branched cover of B 3branched along a tangle.The construction generalizes that of Montesinos [9]and Akbulut and Kirby [1].It is strikingly simple in the case of a two-fold branched cover.We also discuss the related Heegaard decomposition of a p -fold branched cover of an n -tangle.
1Surgery Descriptions
A tangle is a one-manifold properly embedded in a three-ball.An n -tangle is a tangle with 2n boundary points.Let T be an n -tangle and T 0a trivial n -tangle diagram 1(Figure 1).Let D 1∪···∪D n be a disjoint union of disks bounded by T 0and let b 1,...,b m be mutually disjoint disks in
B 3
such that b i ∩ j D j =∂b i ∩T 0are two disjoint arcs in ∂b i (i =1,...,m )(see Figure 2).We denote by Ω(T 0;{D 1,...,D n },{b 1,...,b m })the tangle T 0∪ i ∂b i −int(T 0∩ i ∂b i )and call it a disk-band rep-resentation of a tangle.A disk-band representation is called bicollared if the surfaceseifert
i D i ∪ j b j is orientable.We will see that any n -tangle has a bicollared disk-band representation (Proposition 5).
A framed link is a disjoint union of embedded annuli in a three-manifold.Framed links in S 3can be identified with links whose each component is assigned an integer.Such links are also called framed links .Let M be a three-manifold and L a framed link in M .We denote by Σ(L ,M )the manifold obtained from M by the surgery along L [10].
1Tangles
are considered up to ambient isotopy but in practice we will often use the word tangle for a
tangle diagram or an actual embedding of a one-manifold.
Received by the editors September 15,2001.
AMS subject classification:Primary:57M25;secondary:57M12.Keywords:tangle,branched cover,surgery,Heegaard decomposition.c
Canadian Mathematical Society 2003.356
Branched Covers of Tangles in Three-balls
357
123
n
n −1n −
2
3
Figure 1
Figure 2
The case of two-fold branched covers is easy to visualize so we will formulate it first.
Theorem 1Let Ω(T 0;{D 1,...,D n },{b 1,...,b m })be a disk-band representation of an n-tangle T in B 3.
Let ϕ:H 0→B 3be the two-fold branched cover of B 3by a genus
n −1handlebody H 0branched along T 0.Then the two-fold branched cover of B 3branched along T has a surgery description Σ ϕ−1(
i b i ),H 0 (see Figure
3).
D 3
Ω(T 0;{D 1,...,D n },{b 1,...,b m }
)
D
3
ϕ
(i i 0
Figure 3
Proof Let X be B 3−
i D i compactified with two copies,D ±i ,of D i (i =1,2,...,n )(Figure 4).Let X 1and X 2be two copies of X ,and let D ±
i ,k ⊂X k denote copies of D ±i (i =1,2,...,n ,k =1,2)(Figure 5).Then H 0is obtained from X 1∪X 2by identifying D εi ,1with D −εi ,2(ε∈{−,+}).Let b j ,k =ϕ−1(b j )∩X k and let Y be H 0− j ,k b j ,k compactifi
ed with two copies b ±j ,k of b j ,k in X k (j =1,2,...,m ,k =1,2).Here,+or −sides of D i ,k and b j ,k are not necessarily compatible.We note,and it is the key observation of the construction,that the two-fold branched cover H of B 3branched
along T is obtained from Y by identifying b εj ,1with b −ε
j ,2(ε∈{−,+}).Note that each
b +j ,1∪b −j ,2∪b −j ,1∪b +j ,2is a torus.Let
c j be the core of the annulus b +
j ,1∪b −j ,2.The manifold
obtained from Y by identifying b εj ,1with b −ε
j ,2is homeomorphic to the one obtained
from Y by attaching tori D 2j ×S 1(j =1,2,...,m )so that ∂D 2
j =c j .Hence H is homeomorphic to the manifold with the surgery description Σ ϕ−1(
j b j ),H 0 .
358
Ishiwata,Przytycki and
Yasuhara
Figure 4
Figure 5
Example 2(a)The two-fold branched cover M (2)(T 1)branched along a tangle T 1
in Figure 6is the Seifert manifold with the base a disk and two special fibers of type (2,1)and (2,−1).Furthermore M (2)(T 1)is a twisted I -bundle over the Klein bottle (for example see [7]).In particular,π1 M (2)(T 1)
= a ,b |aba −1b =1 .
(2)(T 1)
D 1
D 2
Figure 6
(b)If we glue together two copies of T 1as in Figure 7,we get Borromean rings L .Thus our previous computation shows that the two-fold branched cover M (2)(L )of S 3branched along L is a “switched”
double of the twisted I -bundle over the Klein bottle (see Figure 8for a surgery description).The fundamental group π1 M (2)
(L ) = x ,a |x 2ax 2a −1,a 2xa 2x −1 is a three-manifold group which is torsion-free but not left orderable [11].
(c)If we take the double of the tangle T 1,we obtain the link in Figure 9.The two-fold branched cover of S 3branched along this link is the double of twisted I -bundle over Klein bottle.A surgery description of this manifold is shown in Figure 10.Thus this manifold is the Seifert manifold with the base S 2and four special fibers of type
Branched Covers of Tangles in Three-balls
359
2−
−
Figure7Figure8Figure9Figure10
a1a2a n−1a n
a1
a n
a n
a1
Figure11Figure12
(2,1),(2,1),(2,−1)and(2,−1).This manifold also has another Seifertfibration, which is a circle bundle over the Klein bottle.
Example2(a)was motivated by the fact that the tangle T1yields a virtual La-grangian of index2in the symplectic space of the Fox Z-colorings of the boundary of our tangle[5].
More generally we have:
Example3Consider a tangle T2in Figure11,called a pretzel tangle of type (a1,a2,...,a n),where each a i is an integer indicating the number of half-twists(i= 1,2,...,n).The two-fold branched cover M(2)(T2)branched along the tangle T2is a Seifertfibered manifold with the base a disk and n specialfibers of type(a1,1),(a2,1), ...,(a n,1)(Figure12).
Theorem1can be generalized to a p-fold cyclic branched cover assuming that an n-tangle is oriented
whose disk-band representation is bicollared,where p is any positive integer greater than2.We proceed as follows:
Let T=Ω(T0;{D1,...,D n},{b1,...,b m})be a bicollared disk-band representa-tion of an n-tangle.Then
i
D i∪
j
b j has a bicollar neighborhood(
i
D i∪
j
b j)×[−1,1].Let X=B3−
(
i
D i)×[−1,1]
and D i±=(D i×[±1,0])∩∂X.Let
X k be a copy of X and D±
i,k
⊂∂X k a copy of D±i(k=1,2,...,p).Then the p-fold cyclic branched coverϕ:H0→B3branched along T0is obtained from X1∪···∪X p
by identifying D+
i,k
with D−
i,k+1
(k=1,...,p),where k is considered modulo p.Let
b±
j,k
=ϕ−1(b j×{±1})∩X k.Note that each b+j,k∪b−
j,k+1
is an annulus in H0for any
360Ishiwata,Przytycki and
Yasuhara
(a)
(b)
(c)
−2
−2
2
2
Figure 13
j and k .Then we obtain the p -fold cyclic branched cover of B 3branched along T in a similar way as in Theorem 1.
Theorem 4Let Ω(T 0;{D 1,...,D n },{b 1,...,b m })be a bicollared disk-band repre-sentation of an n-tangle T in B 3.Then Σ  m j =1  p −1k =1(b +j ,k ∪b −
j ,k +1) ,H 0 is the p-fold cyclic branched cover of B 3branched along T.
Note that we do not use the annuli b +j ,p +1∪b −
j ,1(j =1,2,...,m )in the the-orem above.In fact the cores of these annuli bound mutually disjoint 2-disks in Σ  m j =1  p −1k =1(b +j ,k ∪b −j ,k +1) ,H 0 .
Proof Let Y =H 0− j ϕ−1(b j ×[−1,1]),V ±j ,k =ϕ−1(b j ×[±1,0])∩X k and
β±j ,k =V ±j ,k ∩Y (=∂V ±j ,k ∩∂Y ).Note that ϕ−1
(b j ×[−1,1])is a genus p −1handlebody.Then the p -fold cyclic branched cover of B 3branched along T is home-omorphic to a manifold H that is obtained from Y by identifying β+j ,k with β−
j ,k +1(k =1,...,p ),where k is taken modulo p .Moreover H is homeomorphic to a manifold obtained from H 0− j ϕ−1(b j ×[−1,1])∪ j  V −j ,1
∪V +j ,p ∪ϕ−1(b j ×{0}) by identifying β+j ,k and b j ,k with β−
j ,k +1and b j ,k +1(j =1,...,m ,k =1,...,p −1)re-spectively,where b j ,k =ϕ−1(b j ×{0})∩X k .Note that β+j ,
k ∪b j ,k ∪β−j ,k +1∪b j ,k +1is a torus.By an argument similar to that in the proof of Theorem 1,we have the required result.
Example 5Let T =Ω(T 0;{D },{b 1,b 2})be a tangle as in Figure 13(a)and ϕ:H 0→B 3the three-fold cyclic branched cover of B 3branched along T 0.Note that H 0is a three-ball and ϕ−1(b 1∪b 2)is as shown in Figure 13(b).By Theorem 4,the three-fold cyclic branched cover of B 3branched along T is obtained from H 0by the surgery along a framed link in Figure 13(c).Note that Figure 13(c)is ambient isotopic to Figure 14(a).Since the figure eight knot has a tangle decomposition into T and a trivial 1-tangle,the three-fold cyclic branched cover M (3)(41)of S 3branched along the figure eight knot has a surgery description shown in Figure 14(a).The framed link in Figure 14(a)can be deformed into the link in Figure 14(b)by an ambient isotopy and a second Kirby move.The link in Figure 14(b)is ambient isotopic to the
Branched Covers of Tangles in Three-balls
361
(a)
(b)Figure 14
link in Figure 8.Hence M (3)(41)is homeomorphic to the two-fold branched cover of S 3branched along the Borromean rings [8,6](cf.Example 2(b)).
Proposition 6Any n-tangle (B 3,T )has a (bicollared)disk-band representation.Proof We attach a trivial n -tangle T 0to the n -tangle T by a homeomorphism ϕ:∂(B ,T 0)→∂(B ,T ).We obtain a link L =T 0∪T in a three-sphere B
ϕB with a diagram D (T 0∪T )as in Figure 15.We may assume that the diagram D (T 0∪T )is connected.We color,in checkerboard fashion,the regions of the plane cut by the dia-gram D (T 0∪T )and choose n points {v 1,v 2,...,v n }as in Figure 16.Since D (T 0∪T )is connected,there is a spine G of the black surface with the vertex set V (G )con-taining {v 1,v 2,...,v n }.Deforming G on the surface by edge contractions,we have a new spine H with V (H )={v 1,v 2,...,v n }.By retracting the black regions into the neighborhood of H and restricting to B 3,we have a required surface.For an example,see Figure 17.
When we use the Seifert algorithm instead of checkerboard coloring,we always obtain a bicollared disk-band
representation.
1n −2
1
v 2
v 3
v 1v n −21
2
3
n −2
n −1
n
Figure 15
Figure 16
2Heegaard Decompositions
In addition to the surgery presentation,it is also useful to have another presentation of a p -fold cyclic branched cover.Our construction leads straightforwardly to a Hee-gaard decomposition,that is a decomposition into a compression body [2,3]and a handlebody,of a p -fold cyclic branched cover.

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