On the support genus of a contact structure
ON THE SUPPORT GENUS OF A CONTACT STRUCTURE
MEHMET FIRAT ARIKAN
Abstract. The algorithm given by Akbulut and Ozbagci constructs an explicit open
book decomposition on a contact three-manifold described by a contact surgery on a link
in the three-sphere. In this article, we will improve this algorithm by using Giroux’s con-
tact cell decomposition process. In particular, our algorithm gives a better upper bound
for the recently defined “minimal supporting genus invariant” of contact structures.
1. Introduction
Let (M, ξ) be a closed oriented contact 3-manifold, and let (Σ, h) be an open book (de-
composition) of M which is compatible with the contact structure ξ (sometimes we also
say that (Σ, h) supports ξ). Based on the correspondence theorem (see Theorem 2.3)
between contact structures and their supporting open books, the topological invariant
sg(ξ) was defined in [EO]. More precisely, we have
sg(ξ) = min{ g(Σ) | (Σ, h) an open book decomposition supporting ξ}
called supporting genus of ξ. There are some partial results for this invariant. For instance,
we have:
Theorem 1.1 ([Et1]). If (M, ξ) is overtwisted, then sg(ξ) = 0.
Unlike the overtwisted case, there is not much known yet for sg(ξ) when ξ is tight. On
the other hand, if we, furthermore, require that ξ is Stein fillable, then an algorithm to
find an open book supporting ξ was given in [AO]. Although their construction is explicit,
the pages of the resulting open books arise as Seifert surfaces of torus knots or links, and
so this algorithm is far from even approximating the numbers sg(ξ). In [St], the same
algorithm was generalized to the case where ξ need not to be Stein fillable (or even tight),
but the pages are still of large genera.
This article is organized as follows: After the preliminaries (Section 2), in Section 3
we will present an explicit construction of a supporting open book (with considerably
less genus) for a given contact surgery diagram of any contact structure ξ. Of course,
because of Theorem 1.1, our algorithm makes more sense for the tight structures than
the overtwisted ones. Moreover, it depends on a choice of the contact surgery diagram
describing ξ. Nevertheless, it gives better and more reasonable upper bound for sg(ξ)
(when ξ is tight) as we will see from our examples in Section 4.
Let L be any Legendrian link given in (R3, ξ0 = ker(α0 = dz + xdy)) ⊂ (S
3, ξst). L can
be represented by a special diagram D called a square bridge diagram of L (see [Ly]). We
will consider D as an abstract diagram such that
(1) D consists of horizontal line segments h1, ..., hp, and vertical line segments v1, ..., vq
for some integers p ≥ 2, q ≥ 2,
The author was partially supported by NSF Grant DMS0244622.
http://arxiv.org/abs/0704.1670v4
2 MEHMET FIRAT ARIKAN
(2) there is no collinearity in {h1, . . . , hp}, and in {v1, . . . , vq}.
(3) each hi (resp., each vj) intersects two vertical (resp., horizontal) line segments of
D at its two endpoints (called corners of D), and
(4) any interior intersection (called junction of D) is understood to be a virtual cross-
ing of D where the horizontal line segment is passing over the vertical one.
We depict Legendrian right trefoil and the corresponding D in Figure 1.
Legendrian right trefoil
p = q = 5
Figure 1. The square bridge diagram D for the Legendrian right trefoil
Clearly, for any front projection of a Legendrian link, we can associate a square bridge
diagram D. Using such a diagram D, the following two facts were first proved in [AO],
and later made more explicit in [Pl]. Below versions are from the latter:
Lemma 1.2. Given a Legendrian link L in (R3, ξ0), there exists a torus link Tp,q (with p
and q as above) transverse to ξ0 such that its Seifert surface Fp,q contains L, dα0 is an
area form on Fp,q, and L does not separate Fp,q.
Proposition 1.3. Given L and Fp,q as above, there exist an open book decomposition of
S3 with page Fp,q such that:
(1) the induced contact structure ξ is isotopic to ξ0;
(2) the link L is contained in one of the page Fp,q, and does not separate it;
(3) L is Legendrian with respect to ξ;
(4) there exist an isotopy which fixes L and takes ξ to ξ0, so the Legendrian type of
the link is the same with respect to ξ and ξ0;
(5) the framing of L given by the page Fp,q of the open book is the same as the contact
framing.
Being a Seifert surface of a torus link, Fp,q is of large genera. In Section 3, we will construct
another open book OB supporting (S3, ξst) such that its page F arises as a subsurface of
Fp,q (with considerably less genera), and given Legendrian link L sits on F as how it sits
on the page Fp,q of the construction used in [AO] and [Pl]. The page F of the open book
OB will arise as the ribbon of the 1-skeleton of an appropriate contact cell decomposition
for (S3, ξst). As in [Pl], our construction will keep the given link L Legendrian with respect
to the standard contact structure ξst. Our main theorem is:
Theorem 1.4. Given L and Fp,q as above, there exists a contact cell decomposition ∆ of
(S3, ξst) such that
(1) L is contained in the Legendrian 1-skeleton G of ∆,
ON THE SUPPORT GENUS OF A CONTACT STRUCTURE 3
(2) The ribbon F of the 1-skeleton G is a subsurface of Fp,q (p and q as above),
(3) The framing of L coming from F is equal to its contact framing tb(L), and
(4) If p > 3 and q > 3, then the genus g(F ) of F is strictly less than the genus g(Fp,q)
of Fp,q.
As an immediate consequence (see Corollary 3.1), we get an explicit description of an open
book supporting (S3, ξ) whose page F contains L with the correct framing. Therefore, if
(M±, ξ±) is given by contact (±1)-surgery on L (such a surgery diagram exists for any
closed contact 3-manifold by Theorem 2.1), we get an open book supporting ξ± with page
F by Theorem 2.5. Hence, g(F ) improves the upper bound for sg(ξ) as g(F ) < g(Fp,q)
(for p > 3, q > 3). It will be clear from our examples in Section 4 that this is indeed a
good improvement.
Acknowledgments. The author would like to thank Selman Akbulut, Selahi Durusoy,
Cagri Karakurt, and Burak Ozbagci for their helpful conversations and comments on the
draft of this paper.
2. Preliminaries
2.1. Contact structures and Open book decompositions. A 1-form α ∈ Ω1(M) on
a 3-dimensional oriented manifold M is called a contact form if it satisfies α ∧ dα 6= 0.
An oriented contact structure on M is then a hyperplane field ξ which can be globally
written as the kernel of a contact 1-form α. We will always assume that ξ is a positive
contact structure, that is, α ∧ dα > 0. Note that this is equivalent to asking that dα
be positive definite on the plane field ξ, ie., dα|ξ > 0. Two contact structures ξ0, ξ1 on
a 3-manifold are said to be isotopic if there exists a 1-parameter family ξt (0 ≤ t ≤ 1)
of contact structures joining them. We say that two contact 3-manifolds (M1, ξ1) and
(M2, ξ2) are contactomorphic if there exists a diffeomorphism f : M1 −→ M2 such that
f∗(ξ1) = ξ2. Note that isotopic contact structures give contactomorphic contact manifolds
by Gray’s Theorem. Any contact 3-manifold is locally contactomorphic to (R3, ξ0) where
standard contact structure ξ0 on R
3 with coordinates (x, y, z) is given as the kernel of
α0 = dz + xdy. The standard contact structure ξst on the 3-sphere S
3 = {(r1, r2, θ1, θ2) :
r21 + r
2 = 1} ⊂ C
2 is given as the kernel of αst = r
1dθ1 + r
2dθ2. One basic fact is that
(R3, ξ0) is contactomorphic to (S
3 \ {pt}, ξst). For more details on contact geometry, we
refer the reader to [Ge], [Et3].
An open book decomposition of a closed 3-manifold M is a pair (L, f) where L is an
oriented link in M , called the binding, and f : M \L → S1 is a fibration such that f−1(t)
is the interior of a compact oriented surface Σt ⊂ M and ∂Σt = L for all t ∈ S
1. The
surface Σ = Σt, for any t, is called the page of the open book. The monodromy of an open
book (L, f) is given by the return map of a flow transverse to the pages (all diffeomorphic
to Σ) and meridional near the binding, which is an element h ∈ Aut(Σ, ∂Σ), the group
of (isotopy classes of) diffeomorphisms of Σ which restrict to the identity on ∂Σ . The
group Aut(Σ, ∂Σ) is also said to be the mapping class group of Σ, and denoted by Γ(Σ).
An open book can also be described as follows. First consider the mapping torus
Σ(h) = [0, 1]× Σ/(1, x) ∼ (0, h(x))
where Σ is a compact oriented surface with n = |∂Σ| boundary components and h is an
element of Aut(Σ, ∂Σ) as above. Since h is the identity map on ∂Σ, the boundary ∂Σ(h)
of the mapping torus Σ(h) can be canonically identified with n copies of T 2 = S1 × S1,
4 MEHMET FIRAT ARIKAN
where the first S1 factor is identified with [0, 1]/(0 ∼ 1) and the second one comes from
a component of ∂Σ. Now we glue in n copies of D2 × S1 to cap off Σ(h) so that ∂D2
is identified with S1 = [0, 1]/(0 ∼ 1) and the S1 factor in D2 × S1 is identified with a
boundary component of ∂Σ. Thus we get a closed 3-manifold
M = M(Σ,h) := Σ(h) ∪n D
2 × S1
equipped with an open book decomposition (Σ, h) whose binding is the union of the core
circles in the D2 × S1’s that we glue to Σ(h) to obtain M . To summarize, an element
h ∈ Aut(Σ, ∂Σ) determines a 3-manifold M = M(Σ,h) together with an “abstract” open
book decomposition (Σ, h) on it. For furher details on these subjects, see [Gd], and [Et2].
2.2. Legendrian Knots and Contact Surgery. A Legendrian knot K in a contact
3-manifold (M, ξ) is a knot that is everywhere tangent to ξ. Any Legendrian knot comes
with a canonical contact framing (or Thurston-Bennequin framing), which is defined by
a vector field along K that is transverse to ξ. If K is null-homologous, then this framing
can be given by an integer tb(K), called Thurston-Bennequin number. For any Legendrian
knot K in (R3, ξ0), the number tb(K) can be computed as
tb(K) = bb(K)−#left cusps of K
where bb(K) is the blackboard framing of K.
We call (M, ξ) (or just ξ) overtwisted if it contains an embedded disc D ≈ D2 ⊂ M
with boundary ∂D ≈ S1 a Legendrian knot whose contact framing equals the framing it
receives from the disc D. If no such disc exists, the contact structure ξ is called tight.
For any p, q ∈ Z, a contact (r)-surgery (r = p/q) along a Legendrian knot K in a contact
manifold (M, ξ) was first described in [DG1]. It is defined to be a special kind of a
topological surgery, where surgery coefficient r ∈ Q∪∞ measured relative to the contact
framing of K. For r 6= 0, a contact structure on the surgeried manifold
(M − νK) ∪ (S1 ×D2),
(νK denotes a tubular neighborhood of K) is defined by requiring this contact structure
to coincide with ξ on Y − νK and its extension over S1 × D2 to be tight on (glued in)
solid torus S1 ×D2. Such an extension uniquely exists (up to isotopy) for r = 1/k with
k ∈ Z (see [Ho]). In particular, a contact (±1)-surgery along a Legendrian knot K on a
contact manifold (M, ξ) determines a unique (up to contactomorphism) surgered contact
manifold which will be denoted by (M, ξ)(K,±1).
The most general result along these lines is:
Theorem 2.1 ([DG1]). Every (closed, orientable) contact 3-manifold (M, ξ) can be ob-
tained via contact (±1)-surgery on a Legendrian link in (S3, ξst).
Any closed contact 3-manifold (M, ξ) can be described by a contact surgery diagram. Such
a diagram consists of a front projection (onto the yz-plane) of a Legendrian link drawn in
(R3, ξ0) ⊂ (S
3, ξst) with contact surgery coefficient on each link component. Theorem 2.1
implies that there is a contact surgery diagram for (M, ξ) such that the contact surgery
coefficient of any Legendrian knot in the diagram is ±1. For more details see [Gm] and
[OS].
ON THE SUPPORT GENUS OF A CONTACT STRUCTURE 5
2.3. Compatibility and Stabilization. A contact structure ξ on a 3-manifold M is
said to be supported by an open book (L, f) if ξ is isotopic to a contact structure given by
a 1-form α such that
(1) dα is a positive area form on each page Σ ≈ f−1(pt) of the open book and
(2) α > 0 on L (Recall that L and the pages are oriented.)
When this holds, we also say that the open book (L, f) is compatible with the contact
structure ξ on M . Geometrically, compatibility means that ξ can be isotoped to be
arbitrarily close (as oriented plane fields), on compact subsets of the pages, to the tangent
planes to the pages of the open book in such a way that after some point in the isotopy
the contact planes are transverse to L and transverse to the pages of the open book in a
fixed neighborhood of L.
Definition 2.2. A positive (resp., negative) stabilization S+K(Σ, h) (resp., S
K(Σ, h)) of
an abstract open book (Σ, h) is the open book
(1) with page Σ′ = Σ ∪ 1-handle and
(2) monodromy h′ = h ◦DK (resp., h
′ = h ◦D−1K ) where DK is a right-handed Dehn
twist along a curve K in Σ′ that intersects the co-core of the 1-handle exactly once.
Based on the result of Thurston and Winkelnkemper [TW], Giroux proved the following
theorem which strengthened the link between open books and contact structures.
Theorem 2.3 ([Gi]). Let M be a closed oriented 3-manifold. Then there is a one-to-
one correspondence between oriented contact structures on M up to isotopy and open book
decompositions of M up to positive stabilizations: Two contact structures supported by the
same open book are isotopic, and two open books supporting the same contact structure
have a common positive stabilization.
For a given fixed open book (Σ, h) of a 3-manifold M , there exists a unique compatible
contact structure up to isotopy on M = M(Σ,h) by Theorem 2.3. We will denote this
contact structure by ξ(Σ,h). Therefore, an open book (Σ, h) determines a unique contact
manifold (M(Σ,h), ξ(Σ,h)) up to contactomorphism.
Taking a positive stabilization of an open book (Σ, h) is actually taking a special Murasugi
sum of (Σ, h) with (H+, Dc) where H
+ is the positive Hopf band, and c is the core circle
in H+. Taking a Murasugi sum of two open books corresponds to taking the connect sum
of 3-manifolds associated to the open books. For the precise statements of these facts,
and a proof of the following theorem, we refer the reader to [Gd], [Et2].
Theorem 2.4. (MS+
(Σ,h), ξS+
(Σ,h))
∼= (M(Σ,h), ξ(Σ,h))#(S
3, ξst) ∼= (M(Σ,h), ξ(S,h)).
2.4. Monodromy and Surgery Diagrams. Given a contact surgery diagram for a
closed contact 3-manifold (M, ξ), we want to construct an open book compatible with
ξ. One implication of Theorem 2.1 is that one can obtain such a compatible open book
by starting with a compatible open book of (S3, ξst), and then interpreting the effects of
surgeries (yielding (M, ξ) ) in terms of open books. However, we first have to realize each
surgery curve (in the given surgery diagram of (M, ξ) ) as a Legendrian curve sitting on
a page of some open book supporting (S3, ξst). We refer the reader to Section 5 in [Et2]
for a proof of the following theorem.
6 MEHMET FIRAT ARIKAN
Theorem 2.5. Let (Σ, h) be an open book supporting the contact manifold (M, ξ). If K
is a Legendrian knot on the page Σ of the open book, then
(M, ξ)(K, ±1) = (M(Σ, h◦D∓
), ξ(Σ, h◦D∓
2.5. Contact Cell Decompositions and Convex Surfaces. The exploration of con-
tact cell decompositions in the study of open books was originally initiated by Gabai
[Ga], and then developed by Giroux [Gi]. We want to give several definitions and facts
carefully.
Let (M, ξ) be any contact 3-manifold, and K ⊂ M be a Legendrian knot. The twisting
number tw(K,Fr) of K with respect to a given framing Fr is defined to be the number
of counterclockwise 2π twists of ξ along K, relative to Fr. In particular, if K sits on a
surface Σ ⊂ M , and FrΣ is the surface framing of K given by Σ, then we write tw(K,Σ)
for tw(K,FrΣ). If K = ∂Σ, then we have tw(K,Σ) = tb(K) (by the definition of tb).
Definition 2.6. A contact cell decomposition of a contact 3−manifold (M, ξ) is a finite
CW-decomposition of M such that
(1) the 1-skeleton is a Legendrian graph,
(2) each 2-cell D satisfies tw(∂D,D) = −1, and
(3) ξ is tight when restricted to each 3-cell.
Definition 2.7. Given any Legendrian graph G in (M, ξ), the ribbon of G is a compact
surface R = RG satisfying
(1) R retracts onto G,
(2) TpR = ξp for all p ∈ G,
(3) TpR 6= ξp for all p ∈ R \G.
For a proof of the following lemma we refer the reader to [Gd] and [Et2].
Lemma 2.8. Given a closed contact 3−manifold (M, ξ), the ribbon of the 1-skeleton of
any contact cell decomposition is a page of an open book supporting ξ.
The following lemma will be used in the next section.
Lemma 2.9. Let ∆ be a contact cell decomposition of a closed contact 3-manifold (M, ξ)
with the 1−skeleton G. Let U be a 3-cell in ∆. Consider two Legendrian arcs I ⊂ ∂U
and J ⊂ U such that
(1) I ⊂ G,
(2) J ∩ ∂U = ∂J = ∂I,
(3) C = I ∪∂ J is a Legendrian unknot with tb(C) = −1.
Set G′ = G ∪ J . Then there exists another contact cell decomposition ∆′ of (M, ξ) such
that G′ is the 1-skeleton of ∆′
Proof. The interior of the 3−cell U is contactomorphic to (R3, ξ0). Therefore, there exists
an embedded disk D in U such that ∂D = C and int(D) ⊂ int(U) as depicted in Figure
2(a). We have tw(∂D,D) = −1 since tb(C) = −1. As we are working in (R3, ξ0), there
exist two C∞-small perturbations of D fixing ∂D = C such that perturbed disks intersect
each other only along their common boundary C. In other words, we can find two isotopies
H1, H2 : [0, 1]×D −→ U such that for each i = 1, 2 we have
(1) Hi(t, .) fixes ∂D = C pointwise for all t ∈ [0, 1],
(2) Hi(0, D) = IdD where IdD is the identity map on D,
ON THE SUPPORT GENUS OF A CONTACT STRUCTURE 7
(3) Hi(1, D) = Di where each Di is an embedded disk in U with int(Di) ⊂ int(U),
(4) D ∩D1 ∩D2 = C (see Figure 2(b)).
int(U)
Ext(U)
int(U − U ′)
(a) (b)
Figure 2. Constructing a new contact cell decomposition
Note that tw(∂Di, Di) = tw(C,Di) = −1 for i = 1, 2. This holds because each Di is a
small perturbation of D, so the number of counterclockwise twists of ξ (along K) relative
to FrDi is equal to the one relative to FrD.
Next, we introduce G′ = G ∪ J as the 1-skeleton of the new contact cell decomposition
∆′. In M − int(U), we define the 2- and 3- skeletons of ∆′ to be those of ∆ . However,
we change the cell structure of int(U) as follows: We add 2-cells D1, D2 to the 2-skeleton
of ∆′ (note that they both satisfy the twisting condition in Definition 2.6). Consider the
2-sphere S = D1 ∪D2 where the union is taken along the common boundary C. Let U
be the 3-ball with ∂U ′ = S. Note that ξ|U ′ is tight as U
′ ⊂ U and ξ|U is tight. We add U ′
and U −U ′ to the 3-skeleton of ∆′ (note that U −U ′ can be considered as a 3-cell because
observe that int(U − U ′) is homeomorphic to the interior of a 3-ball as in Figure 2(b)).
Hence, we established another contact cell decomposition of (M, ξ) whose 1-skeleton is
G′ = G∪J . (Equivalently, by Theorem 2.4, we are taking the connect sum of (M, ξ) with
(S3, ξst) along U
′.) �
3. The Algorithm
3.1. Proof of Theorem 1.4.
Proof. By translating L in (R3, ξ0) if necessary (without changing its contact type), we
can assume that the front projection of L onto the yz-plane lying in the second quadrant
{ (y, z) | y < 0, z > 0}. After an appropriate Legendrian isotopy, we can assume that L
consists of the line segments contained in the lines
ki = {x = 1, z = −y + ai}, i = 1, . . . , p,
lj = {x = −1, z = y + bj}, j = 1, . . . , q
for some a1 < a2 < · · · < ap, 0 < b1 < b2 < · · · < bq, and also the line segments (parallel
to the x-axis) joining certain ki’s to certain lj ’s. In this representation, L seems to have
8 MEHMET FIRAT ARIKAN
corners. However, any corner of L can be made smooth by a Legendrian isotopy changing
only a very small neighborhood of that corner.
Let π : R3 −→ R2 be the projection onto the yz-plane. Then we obtain the square bridge
diagram D = π(L) of L such that D consists of the line segments
hi ⊂ π(ki) = {x = 0, z = −y + ai}, i = 1, . . . , p,
vj ⊂ π(lj) = {x = 0, z = y + bj}, j = 1, . . . , q.
Notice that D bounds a polygonal region P in the second quadrant of the yz-plane, and
divides it into finitely many polygonal subregions P1, . . . , Pm ( see Figure 3-(a) ).
Throughout the proof, we will assume that the link L is not split (that is, the region P
has only one connected component). Such a restriction on L will not affect the generality
of our construction (see Remark 3.2).
a1 a2 a3 a4 a5
P2 P3
h2 h3
Figure 3. The region P for right trefoil knot and its division into rectangles
Now we decompose P into finite number of ordered rectangular subregions as follows:
The collection {π(lj) | j = 1, . . . , q} cuts each Pk into finitely many rectangular regions
R1k, . . . , R
k . Consider the set P of all such rectangles in P . That is, we define
= { Rlk | k = 1, . . . , m, l = 1, . . . , mk}.
Clearly P decomposes P into rectangular regions ( see Figure 3-(b) ). The boundary of
an arbitrary element Rlk in P consists of four edges: Two of them are the subsets of the
lines π(lj(k,l)), π(lj(k,l)+1), and the other two are the subsets of the line segments hi1(k,l),
hi2(k,l) where 1 ≤ i1(k, l) < i2(k, l) ≤ p and 1 ≤ j(k, l) < j(k, l) + 1 ≤ q (see Figure 4).
Since the region P has one connected component, the following holds for the set P:
ON THE SUPPORT GENUS OF A CONTACT STRUCTURE 9
π(lj(k,l))
π(lj(k,l)+1)
hi1(k,l)
hi2(k,l)
ai1(k,l) ai2(k,l)
bj(k,l)
bj(k,l)+1
Figure 4. Arbitrary element Rlk in P
(⋆) Any element of P has at least one common vertex with some other element of P.
By (⋆), we can rename the elements of P by putting some order on them so that any
element of P has at least one vertex in common with the union of all rectangles coming
before itself with respect to the chosen order. More precisely, we can write
P = { Rk | k = 1, . . . , N}
(N is the total number of rectangles in P) such that each Rk has at least one vertex in
common with the union R1 ∪ · · · ∪ Rk−1.
Equivalently, we can construct the polygonal region P by introducing the building rect-
angles (Rk’s) one by one in the order given by the index set {1, 2, . . . , N}. In particular,
this eliminates one of the indexes, i.e., we can use Rk’s instead of R
k’s. In Figure 5, how
we build P is depicted for the right trefoil knot (compare it with the previous picture
given for P in Figure 3-(b)).
π(k1)
π(l5)
π(l1)
Figure 5. The region P for right trefoil knot
10 MEHMET FIRAT ARIKAN
Using the representation P = R1 ∪ R2 ∪ · · · ∪ RN , we will construct the contact cell
decomposition (CCD) ∆. Consider the following infinite strips which are parallel to the
x-axis (they can be considered as the unions of “small” contact planes along ki’s and lj ’s):
S+i = {1− ǫ ≤ x ≤ 1 + ǫ, z = y + ai}, i = 1, . . . , p,
S−j = {−1− ǫ ≤ x ≤ −1 + ǫ, z = −y + bj}, j = 1, . . . , q.
Note that π(S+i ) = π(ki) and π(S
j ) = π(lj). Let Rk ⊂ P be given. Then we can write
∂Rk = C
k ∪ C
k ∪ C
k ∪ C
k where C
k ⊂ π(ki1), C
k ⊂ π(lj), C
k ⊂ π(ki2), C
k ⊂ π(lj+1)
for some 1 ≤ i1 < i2 ≤ p and 1 ≤ j ≤ q. Lift C
k , C
k , C
k , C
k (along the x-axis) so that
the resulting lifts (which will be denoted by the same letters) are disjoint Legendrian arcs
contained in ki1 , lj, ki2, lj+1 and sitting on the corresponding strips S
, S−j , S
, S−j+1. For
l = 1, 2, 3, 4, consider Legendrian linear arcs I lk (parallel to the x-axis) running between
the endpoints of C lk’s as in Figure 6-(a)&(b). Along each I
k the contact planes make a 90
left-twist. Let Blk be the narrow band obtained by following the contact planes along I
Then define Fk to be the surface constructed by taking the union of the compact subsets
of the above strips (containing corresponding C lk’s) with the bands B
k’s (see Figure 6-(b)).
C lk’s and I
k’s together build a Legendrian unknot γk in (R
3, ξ0), i.e., we set
γk = C
k ∪ I
k ∪ C
k ∪ I
k ∪ C
k ∪ I
k ∪ C
k ∪ I
Note that π(γk) = ∂Rk, γk sits on the surface Fk, and Fk deformation retracts onto γk.
Indeed, by taking all strips and bands in the construction small enough, we may assume
that contact planes are tangent to the surface Fk only along the core circle γk. Thus, Fk
is the ribbon of γk. Observe that, topologically, Fk is a positive (left-handed) Hopf band.
(a) (b) (c)
0−1 1
Dk ≈ fk(Rk)
Figure 6. (a) The Legendrian unknot γk, (b) The ribbon Fk, (c) The disk
Dk (shaded bands in (b) are the bands B
Let fk : Rk −→ R
3 be a function modelled by (a, b) 7→ c = a2 − b2 (for an appropriate
choice of coordinates). The image fk(Rk) is, topologically, a disk, and a compact subset
of a saddle surface. Deform fk(Rk) to another “saddle” disk Dk such that ∂Dk = γk (see
Figure 6-(c)). We observe here that tw(γk, Dk) = −1 because along γk, contact planes
rotate 90◦ in the counter-clockwise direction exactly four times which makes one full left-
twist (enough to count the twists of the ribbon Fk since Fk rotates with the contact planes
along γk !).
ON THE SUPPORT GENUS OF A CONTACT STRUCTURE 11
We repeat the above process for each rectangle Rk in P and get the set
D = { Dk | Dk ≈ fk(Rk), k = 1, . . . , N}
consisting of the saddle disks. Note that by the construction of D, we have the property:
(∗) If any two elements of D intersect each other, then they must intersect along a
contractible subset (a contractible union of linear arcs) of their boundaries.
For instance, if the corresponding two rectangles (for two intersecting disks in D) have
only one common vertex, then those disks intersect each other along the (contractible)
line segment parallel to the x-axis which is projected (by the map π) onto that vertex.
For each k, let D′k be a disk constructed by perturbing Dk slightly by an isotopy fixing
only the boundary of Dk. Therefore, we have
(∗∗) ∂Dk = γk = ∂D
k , int(Dk) ∩ int(D
k) = ∅ , and tw(γk, D
k) = −1 = tw(γk, Dk).
In the following, we will define a sequence { ∆k | k = 1, . . . , N } of CCD’s for (S
3, ξst).
∆1k,∆
k, and ∆
k will denote the 1-skeleton, 2-skeleton, and 3-skeleton of ∆k, respectively.
First, take ∆11 = γ1, and ∆
1 = D1 ∪γ1 D
1. By (∗∗), ∆1 satisfies the conditions (1) and
(2) of Definition 2.6. By the construction, any pair of disks Dk, D
k (together) bounds
a Darboux ball (tight 3-cell) Uk in the tight manifold (R
3, ξ0). Therefore, if we take
∆31 = U1 ∪∂ (S
3 − U1), we also achieve the condition (3) in Definition 2.6 ( the boundary
union “ ∪∂” is taken along ∂U1 = S
2 = ∂(S3 − U1) ). Thus, ∆1 is a CCD for (S
3, ξst).
Inductively, we define ∆k from ∆k−1 by setting
∆1k = ∆
k−1 ∪ γk = γ1 ∪ · · · ∪ γk−1 ∪ γk,
∆2k = ∆
k−1 ∪Dk ∪γk D
k = D1 ∪γ1 D
1 ∪ · · · ∪Dk−1 ∪γk−1 D
k−1 ∪Dk ∪γk D
∆3k = U1 ∪ · · · ∪ Uk−1 ∪ Uk ∪∂ (S
3 − U1 ∪ · · · ∪ Uk−1 ∪ Uk)
Actually, at each step of the induction, we are applying Lemma 2.9 to ∆k−1 to get ∆k.
We should make several remarks: First, by the construction of γk’s, the set
(γ1 ∪ · · · ∪ γk−1) ∩ γk
is a contractible union of finitely many arcs. Therefore, the union ∆1k−1 ∪ γk should be
understood to be a set-theoretical union (not a topological gluing!) which means that
we are attaching only the (connected) part (γk \ ∆
k−1) of γk to construct the new 1-
skeleton ∆1k. In terms of the language of Lemma 2.9, we are setting I = ∆
k−1 \ γk and
J = γk \∆
k−1. Secondly, we have to show that ∆
k = ∆
k−1 ∪Dk ∪γk D
k can be realized
as the 2-skeleton of a CCD: Inductively, we can achieve the twisting condition on 2-cells
by using (∗∗). The fact that any two intersecting 2-cells in ∆2k intersect each other along
some subset of the 1-skeleton ∆1k is guaranteed by the property (∗) if they have different
index numbers, and guaranteed by (∗∗) if they are of the same index. Thirdly, we have
to guarantee that 3-cells meet correctly: It is clear that U1, . . . , Uk meet with each other
along subsets of the 1-skeleton ∆1k(⊂ ∆
k). Observe that ∂(U1 ∪ · · · ∪ Uk) = S
2 for any
k = 1, . . . , N by (∗) and (∗∗). Therefore, we can always consider the complementary
Darboux ball S3 − U1 ∪ · · · ∪ Uk−1 ∪ Uk, and glue it to U1 ∪ · · · ∪ Uk along their common
boundary 2-sphere. Hence, we have seen that ∆k is a CCD for (S
3, ξst) with Legendrian
1-skeleton ∆1k = γ1 ∪ · · · ∪ γk.
12 MEHMET FIRAT ARIKAN
To understand the ribbon, say Σk, of ∆
k, observe that when we glue the part γk \∆
k−1 of
γk to ∆
k−1, actually we are attaching a 1-handle (whose core interval is (γk \∆
k−1)\Σk−1)
to the old ribbon Σk−1 (indeed, this corresponds to a positive stabilization). We choose
the 1-handle in such a way that it also rotates with the contact planes. This is equivalent
to extending Σk−1 to a new surface by attaching the missing part (the part which retracts
onto (γk \∆
k−1) \ Σk−1) of Fk given in Figure 6-(c). The new surface is the ribbon Σk of
the new 1-skeleton ∆1k.
By taking k = N , we get a CCD ∆N of (S
3, ξst). By the construction, γk’s are only
piecewise smooth. We need a smooth embedding of L into the 1-skeleton ∆1N (the union
of all γk’s). Away from some small neighborhood of the common corners of ∆
N and L
(recall that L had corners before the Legendrian isotopies), L is smoothly embedded in
∆1N . Around any common corner, we slightly perturb ∆
N using the isotopy used for
smoothing that corner of L. This guaranties the smooth Legendrian embedding of L into
the Legendrian graph ∆1N = ∪
k=1γk. Similarly, any other corner in ∆
N (which is not in
L) can be made smooth using an appropriate Legendrian isotopy.
As L is contained in the 1-skeleton ∆1N , L sits (as a smooth Legendrian link) on the ribbon
ΣN . Note that during the process we do not change the contact type of L, so the contact
(Thurston-Bennequin) framing of L is still the same as what it was at the beginning. On
the other hand, consider tubular neighborhood N(L) of L in ΣN . Being a subsurface
of the ribbon ΣN , N(L) is the ribbon of L. By definition, the contact framing of any
component of L is the one coming from the ribbon of that component. Therefore, the
contact framing and the N(L)-framing of L are the same. Since N(L) ⊂ ΣN , the framing
which L gets from the ribbon ΣN is the same as the contact framing of L. Finally, we
observe that ΣN is a subsurface of the Seifert surface Fp,q of the torus link (or knot) Tp,q.
To see this, note that P is contained in the rectangular region, say Pp,q, enclosed by the
lines π(k1), π(kp), π(l1), π(lq). Divide Pp,q into the rectangular subregions using the lines
π(ki), π(lj), i = 1, . . . , p, j = 1, . . . , q. Note that there are exactly pq rectangles in the
division. If we repeat the above process using this division of Pp,q, we get another CCD
for (S3, ξst) with the ribbon Fp,q. Clearly, Fp,q contains our ribbon ΣN as a subsurface
(indeed, there are extra bands and parts of strips in Fp,q which are not in ΣN ).
Thus, (1), (2) and (3) of the theorem are proved once we set ∆ = ∆N , (and so G = ∆
F = ΣN ). To prove (4), recall that we are assuming p > 3, q > 3. Then consider
= total number of intersection points of all π(lj)’s with all hi’s.
That is, we define κ
= |{π(lj) | j = 1, . . . , q} ∩ {hi | i = 1, . . . , p} |. Notice that κ is the
number of bands used in the construction of the ribbon F , and also that if D (so P ) is
not a single rectangle (equivalently p > 2, q > 2), then κ < pq. Since there are p+ q disks
in F , we compute the Euler characteristic and genus of F as
χ(F ) = p+ q − κ = 2− 2g(F )− |∂F | =⇒ g(F ) =
2− p− q
|∂F |
Similarly, there are p + q disks and pq bands in Fp,q, so we get
χ(Fp,q) = p+ q − pq = 2− 2g(Fp,q)− |∂Fp,q| =⇒ g(Fp,q) =
2− p− q
|∂Fp,q|
ON THE SUPPORT GENUS OF A CONTACT STRUCTURE 13
Observe that |∂Fp,q| divides the greatest common divisor gcd(p, q) of p and q, so
|∂Fp,q| ≤ gcd(p, q) ≤ p =⇒ g(Fp,q) ≥
2− p− q
Therefore, to conclude g(F ) < g(Fp,q), it suffices to show that pq−κ > p−|∂F |. To show
the latter, we will show pq − κ− p ≥ 0 (this will be enough since |∂F | 6= 0).
Observe that pq−κ is the number of bands (along x-axis) in Fp,q which we omit to get the
ribbon F . Therefore, we need to see that at least p bands are omitted in the construction
of F : The set of all bands (along x-axis) in Fp,q corresponds to the set
{π(lj) | j = 1, . . . , q} ∩ {π(ki) | i = 1, . . . , p}.
Notice that while constructing F we omit at least 2 bands corresponding to the intersec-
tions of the lines π(k1), π(kp) with the family {π(lj) | j = 1, . . . , q} (in some cases, one of
these bands might correspond to the intersection of the lines π(k2) or π(kp−1) with π(l1)
or π(lq), but the following argument still works because in such a case we can omit at
least 2 bands corresponding to two points on π(k2) or π(kp−1)). For the remaining p− 2
line segments h2, . . . , hp−1, there are two cases: Either each hi, for i = 2, . . . , p− 1 has at
least one endpoint contained on a line other than π(l1) or π(lq), or there exists a unique
hi, 1 < i < p, such that its endpoints are on π(l1) and π(lq) (such an hi must be unique
since no two vj ’s are collinear !). If the first holds, then that endpoint corresponds to the
intersection of hi with π(lj) for some j 6= 1, q. Then the band corresponding to either
π(ki)∩π(lj−1) or π(ki)∩π(lj+1) is omitted in the construction of F (recall how we divide
P into rectangular regions). If the second holds, then there is at least one line segment
hi′ , which belongs to the same component of L containing hi, such that we omit at least 2
points on π(ki′) (this is true again since no two vj ’s are collinear). Hence, in any case, we
omit at least p bands from Fp,q to get F . This completes the proof of Theorem 1.4. �
Corollary 3.1. Given L and Fp,q as in Theorem 1.4, there exists an open book decompo-
sition OB of (S3, ξst) such that
(1) L lies (as a Legendrian link) on a page F of OB,
(2) The page F is a subsurface of Fp,q
(3) The page framing of L coming from F is equal to its contact framing tb(L),
(4) If p > 3 and q > 3, then g(F ) is strictly less than g(Fp,q),
(5) The monodromy h of OB is given by h = tγ1 ◦ · · · ◦ tγN where γk is the Legendrian
unknot constructed in the proof of Theorem 1.4, and tγk denotes the positive (right-
handed) Dehn twist along γk.
Proof. The proofs of (1), (2), (3), and (4) immediately follow from Theorem 1.4 and
Lemma 2.8. To prove (5), observe that by adding the missing part of each γk to the
previous 1-skeleton, and by extending the previous ribbon by attaching the ribbon of the
missing part of γk (which is topologically a 1-handle), we actually positively stabilize the
old ribbon with the positive Hopf band (H+, tγk). Therefore, (5) follows. �
With a little more care, sometimes we can decrease the number of 2-cells in the final
2-skeleton. Also the algorithm can be modified for split links:
Remark 3.2. Under the notation used in the proof of Theorem 1.4, we have the following:
(1) Suppose that the link L is split (so P has at least two connected components).
Then we can modify the above algorithm so that Theorem 1.4 still holds.
14 MEHMET FIRAT ARIKAN
(2) Let Tj denote the row (or set) of rectangles (or elements) in P (or in P) with
bottom edges lying on the fixed line π(lj). Consider two consecutive rows Tj , Tj+1
lying between the lines π(lj), π(lj+1), and π(lj+2). Let R ∈ Tj and R
′ ⊂ Tj+1 be
two rectangles in P with boundaries given as
∂R = C1 ∪ C2 ∪ C3 ∪ C4, ∂R
′ = C ′1 ∪ C
2 ∪ C
3 ∪ C
Suppose that R and R′ have one common boundary component lying on π(lj+1),
and two of the other components lie on the same lines π(ki1), π(ki2) as in Figure
7. Let γ, γ′ ⊂ ∆1N and D,D
′ ⊂ ∆N be the corresponding Legendrian unknots and
2-cells of the CCD ∆N coming from R,R
′. That is,
∂D = γ, ∂D′ = γ′, and π(D) = R, π(D′) = R′
Suppose also that L∩ γ ∩ γ′ = ∅. Then in the construction of ∆N , we can replace
R,R′ ⊂ P with a single rectangle R′′ = R∪R′. Equivalently, we can take out γ∩γ′
from ∆1N , and replace D,D
′ by a single saddle disk D′′ with ∂D′′ = (γ∪γ′)\(γ∩γ′).
π(lj)
π(lj+1)
π(ki1)
π(ki2)
ai1 ai2
R′′ C1
π(lj+2)
Figure 7. Replacing R, R′ with their union R′′
Proof. To prove each statement, we need to show that CCD structure and all the conclu-
sions in Theorem 1.4 are preserved after changing ∆N the way described in the statement.
To prove (1), let P (1), . . . , P (m) be the separate components of P . After putting the
corresponding separate components of L into appropriate positions (without changing
their contact type) in (R3, ξ0), we may assume that the projection
P = P (1) ∪ · · · ∪ P (m)
of L onto the second quadrant of the yz-plane is given similar as the one which we
illustrated in Figure 8.
In such a projection, we require two important properties:
(1) P (1), . . . , P (m) are located from left to right in the given order in the region bounded
by the lines π(k1), π(l1), and π(lq).
(2) Each of P (1), . . . , P (m) has at least one edge on the line π(l1).
ON THE SUPPORT GENUS OF A CONTACT STRUCTURE 15
π(lq)
π(l1)
π(k1)
(m−1)
Figure 8. Modifying the algorithm for the case when L is split
If the components P (1) . . . P (m) remain separate, then our construction in Theorem 1.4
cannot work (the complement of the union of 3-cells corresponding to the rectangles in
P would not be a Darboux ball; it would be a genus m handle body). So we have
to make sure that any component P (l) is connected to the some other via some bridge
consisting of rectangles. We choose only one rectangle for each bridge as follows: Let
Al be the rectangle in T1 (the row between π(l1) and π(l2)) connecting P
(l) to P (l+1)
for l = 1, . . . , m − 1 (see Figure 8). Now, by adding 2- and 3-cells (corresponding to
A1, . . . , Am−1), we can extend the CCD ∆N to get another CCD for (S
3, ξst). Therefore,
we have modified our construction when L is split.
To prove (2), if we replace D′′ in the way described above, then by the construction of
∆3N , we also replace two 3-cells with a single 3-cell whose boundary is the union of D
and its isotopic copy. This alteration of ∆3N does not change the fact that the boundary
of the union of all 3-cells coming from all pairs of saddle disks is still homeomorphic to a
2-sphere S2, Therefore, we can still complete this union to S3 by gluing a complementary
Darboux ball. Thus, we still have a CCD. Note that γ ∩ γ′ is taken away from the 1-
skeleton. However, since L∩ γ ∩ γ′ = ∅, the new 1-skeleton still contains L. Observe also
that this process does not change the ribbon N(L) of L. Hence, the same conclusions in
Theorem 1.4 are satisfied by the new CCD. �
16 MEHMET FIRAT ARIKAN
4. Examples
Example I. As the first example, let us finish the one which we have already started
in the previous section. Consider the Legendrian right trefoil knot L (Figure 1) and the
corresponding region P given in Figure 5. Then we construct the 1-skeleton, the saddle
disks, and the ribbon of the CCD ∆ as in Figure 9.
All twists are left-handed
Figure 9. (a) The page F for the right trefoil knot, (b) Construction of ∆
ON THE SUPPORT GENUS OF A CONTACT STRUCTURE 17
In Figure 9-(a), we show how to construct the 1-skeleton G = ∆1 of ∆ starting from a
single Legendrian arc (labelled by the number “ 0 ”). We add Legendrian arcs labelled
by the pairs of numbers “1, 1”, . . . ,“8, 8” to the picture one by one (in this order). Each
pair determines the endpoints of the corresponding arc. These arcs represent the cores
of the 1-handles building the page F (the ribbon of G) of the corresponding open book
OB. Note that by attaching each 1-handle, we (positively) stabilize the previous ribbon
by the positive Hopf band (H+
, tγk) where γk is the boundary of the saddle disk Dk as
before. Therefore, the monodromy h of OB supporting (S3, ξst) is given by
h = tγ1 ◦ · · · ◦ tγ8
where tγk ∈ Aut(F, ∂F ) denotes the positive (right-handed) Dehn twist along γk. To
compute the genus gF of F , observe that F is constructed by attaching eight 1-handles
(bands) to a disk, and |∂F | = 3 where |∂F | is the number of boundary components of F .
Therefore,
χ(F ) = 1− 8 = 2− 2gF − |∂F | =⇒ gF = 3.
Now suppose that (M±1 , ξ
1 ) is obtained by performing contact (±1)-surgery on L. Clearly,
the trefoil knot L sits as a Legendrian curve on F by our construction, so by Theorem
2.5, we get the open book (F, h1) supporting ξ with monodromy
h1 = tγ1 ◦ · · · ◦ tγ8 ◦ t
L ∈ Aut(F, ∂F ).
Hence, we get an upper bound for the support genus invariant of ξ1, namely,
sg(ξ1) ≤ 3 = gF .
We note that the upper bound, which we can get for this particular case, from [AO] and
[St] is 6 where the page of the open book is the Seifert surface F5,5 of the (5, 5)-torus link
(see Figure 10).
z + y = 0x
z − y = 0
All twists are left-handed
Figure 10. Legendrian right trefoil knot sitting on F5,5
Example II. Consider the Legendrian figure-eight knot L, and its square bridge position
given in Figure 11-(a) and (b). We get the corresponding region P in Figure 11-(c). Using
Remark 3.2 we replace R5 and R8 with a single saddle disk. So this changes the set P.
Reindexing the rectangles in P, we get the decomposition in Figure 12 which will be used
to construct the CCD ∆.
18 MEHMET FIRAT ARIKAN
a6a5a4a3a2a1
Figure 11. (a),(b) Legendrian figure-eight knot, (c) The region P
π(l1)
π(l6)
π(k1)
Figure 12. Modifying the region P
In Figure 13-(a), similar to Example I, we construct the 1-skeleton G = ∆1 of ∆ again by
attaching Legendrian arcs (labelled by the pairs of numbers “1, 1”, . . . , “10, 10”) to the
initial arc (labelled by the number “0”) in the given order. Again each pair determines
the endpoints of the corresponding arc, and the cores of the 1-handles building the page F
(of the corresponding open book OB). Once again attaching each 1-handle is equivalent
to (positively) stabilizing the previous ribbon by the positive Hopf band (H+
, tγk) for
k = 1, . . . , 10. Therefore, the monodromy h of OB supporting (S3, ξst) is given by
h = tγ1 ◦ · · · ◦ tγ10
To compute the genus gF of F , observe that F is constructed by attaching ten 1-handles
(bands) to a disk, and |∂F | = 5. Therefore,
χ(F ) = 1− 10 = 2− 2gF − |∂F | =⇒ gF = 3.
ON THE SUPPORT GENUS OF A CONTACT STRUCTURE 19
D10(b)
z + y = 0
z − y = 0
All twists are left-handed
All twists are left-handed
Figure 13. (a) The page F , (b) Construction of ∆, (c) The figure-eight
knot on F6,6
20 MEHMET FIRAT ARIKAN
Let (M±2 , ξ
2 ) be a contact manifold obtained by performing contact (±)-surgery on the
figure-8 knot L. Since L sits as a Legendrian curve on F by our construction, Theorem
2.5 gives an open book (F, h2) supporting ξ2 with monodromy
h2 = tγ1 ◦ · · · ◦ tγ10 ◦ t
L ∈ Aut(F, ∂F ).
Therefore, we get the upper bound sg(ξ2) ≤ 3 = gF . Once again we note that the smallest
possible upper bound, which we can get for this particular case, using the method of [AO]
and [St] is 10 where the page of the open book is the Seifert surface F6,6 of the (6, 6)-torus
link (see Figure 13-(c)).
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Department of Mathematics, MSU, East Lansing MI 48824, USA
E-mail address : arikanme@msu.edu
http://arxiv.org/abs/math/0605441
	1. Introduction
	2. Preliminaries
	2.1. Contact structures and Open book decompositions
	2.2. Legendrian Knots and Contact Surgery 
	2.3. Compatibility and Stabilization
	2.4. Monodromy and Surgery Diagrams
	2.5. Contact Cell Decompositions and Convex Surfaces
	3. The Algorithm
	3.1. Proof of Theorem ??
	4. Examples
	References
