On the support genus of Legendrian knots

Abstract

In this paper, we show that any topological knot or link in $S^1 \times S^2$ sits on a planar page of an open book decomposition whose monodromy is a product of positive Dehn twists. As a consequence, any knot or link type in $S^1 \times S^2$ has a Legendrian representative having support genus zero. We also show this holds for some knots and links in the lens spaces L(p, 1).

1 Introduction

In 2002, Giroux showed that given any contact structure $\xi$ on a 3-manifold M, there is an open book decomposition of the 3-manifold such that the contact structure is transverse to the binding of the open book and it can be isotoped arbitrarily close to the pages. In this case, we say the contact 3-manifold $(M, \xi )$ is supported by this open book decomposition. Further, Giroux showed that a contact 3-manifold is Stein fillable (and hence tight) if and only if there is an open book decomposition for the contact 3-manifold whose monodromy can be written as a product of positive Dehn twists, [5]. The purpose of this note is to construct the simplest open book decomposition of lens spaces whose monodromy is a product of positive Dehn twists and which contains a given knot or link on its page. The first result explicitly constructs such open book decomposition for knots and links in $S^1 \times S^2$.

Theorem 1.1

Any topological knot or link in $S^1 \times S^2$ sits on a planar page of an open book decomposition whose monodromy is a product of positive Dehn twists.

Similar to the support genus invariant of a contact structure defined in [3], a new invariant of a Legendrian knot L in a contact 3-manifold $(M, \xi )$ is defined using open book decompositions in [7]. The invariant is called the support genus sg(L) of L and it is defined as the minimal genus of a page of an open book decomposition of M supporting $\xi$ such that L sits on a page of the open book and the framings given by $\xi$ and the page agree. In [7], it was shown that while any null-homologous knot with an overtwisted complement has $sg(L) = 0$, not all Legendrian knots have. Moreover, it was shown that any knot or link in the 3-sphere $S^3$ sits on a planar page of an open book decomposition whose monodromy is a product of positive Dehn twists. This guarantees a Legendrian representative of the given knot or link having support genus zero in the standard tight contact $S^3$. This paper addresses this theme for lens spaces L(p, 1).

According to [2], $S^1 \times S^2$ has a unique tight and Stein fillable contact structure. Then by [5], the open book decomposition constructed in Theorem 1.1 supports this unique tight and Stein fillable contact structure. As a consequence of Theorem 1.1, we have the following result on the support genus of Legendrian knots or links in $S^1 \times S^2$.

Theorem 1.2

Given a knot or link type K in the tight contact $S^1 \times S^2$, there is a Legendrian representative L of K such that $sg(L) = 0$.

The proof of Theorem 1.1 does not extend to lens spaces (see Remark 2.1); however, using another method, we show the following.

Theorem 1.3

Let K be a knot or link type in $L(p, 1)$ which is presented as a 2n-pure braided plat as in Fig. 2, $n \in {\mathbb {N}}$. For $p > 2n - 2$, there is a tight contact structure $\xi$ on $L(p, 1)$ and a Legendrian representative L of K in $(L(p, 1), \xi )$ such that $sg(L) = 0$.

Remark 1.4

Note that Theorem 1.1 generalizes to knots or links in the connected sums $\#_m S^1\times S^2$ and gives Theorem 1.5 below. This generalization is handy in studying contact geometry in dimension 5. In [1], Acu, Etnyre, and Ozbagci use this result to study open book decompositions of 5-manifolds and show that many sub-critically fillable contact 5-manifolds are supported by iterated planar open book decompositions. Theorem 1.3 generalises to $\#_mL(p, 1)$ as well.

Theorem 1.5

Let L be a topological link in $\#_m S^1\times S^2$. Then there is a planar open book decomposition for $\#_m S^1\times S^2$ supporting the unique tight contact structure on this manifold such that L can be realized as a Legendrian link on a page of the open book.

In the following section, we prove the main theorems and discuss examples. We conclude with a handful of questions related to the support genus of Legendrian knots in Section 3. The reader may refer to [4] for background information on contact structures.

2 Knots and links in L (p, 1)

The lens space $L(p, 1)$ can be obtained by a surgery along a single $(-p)$-framed unknot U in $S^3$. Given any knot K in $L(p, 1)$, take a projection of K in the surgery diagram for $L(p, 1)$ such that the unknot U intersects the plane of the projection transversely at a single point. The knot K in the surgery diagram of $L(p, 1)$ can be given as in Fig. 1 which is called a mixed link diagram of the link $K \cup U$. When $p=0$, surgery along the 0-framed unknot U gives $S^1 \times S^2$.

Fig. 1

Knots in the surgery diagram of $L(p, 1)$

After isotoping the $(-p)$-framed unknot as in Fig. 2, we may present the knot K as a pure braided plat by using [7, Lemma 3.3]. The pure braided plat of K is a shifted 2n-plat given in Fig. 2 where 2n-braid is a pure braid.

Fig. 2

Pure braided plat presentation of K in $L(p, 1)$

Given a knot K in $L(p, 1)$ as in Fig. 2, we decompose K and the isotoped unknot U in terms of the standard generators of the pure braid group. See Fig. 3 for a decomposition of U.

Fig. 3

A decomposition of the surgery unknot U

Proof of Theorem 1.1

We start with a decomposition of the knot K and the isotoped unknot U in $S^1\times S^2$ in terms of the standard generators of the pure braid group. The link diagram of $K\cup U$, in this case, contains only full twists. We can remove these full twists by blowing up, by adding an unknot with framing $\pm 1$ which changes the framings by adding $\pm 1$. We remove the full twists of U as in Fig. 4 so that U has linking number 1 with K. Next we continue blowing up at the points indicated in Fig. 4 and the surgery framing of U is 0 again, see Fig. 5.

Fig. 4

Removing the full twist by blowing up

Fig. 5

The unknot U has linking number 1 with K and it has framing 0

Then we unknot K by following the algorithm from [7, Remark 5.1 and Theorem 5.2] that puts the knots in $S^3$ on a planar page of an open book decomposition whose monodromy is a product of positive Dehn twists. The unknotted K (we call it $U_K$) bounds a disk in $S^3$ and $S^3$ has a natural open book decomposition with this disk page and with the monodromy $\phi = Id$. We will use this open book decomposition to construct the desired open book of $S^1 \times S^2$ containing K on its page. A page of an open book decomposition of $S^1 \times S^2$ is obtained from the disk bounded by $U_K$ after 0-framed surgeries. More specifically, each 0-framed unknot (in particular, the unknot U) is arranged to puncture each disk page transversely once to form a binding component after surgery. Each $-1$-framed unknot is arranged to be isotoped onto one of the punctured disk pages to perform the surgery on the page. Performing $-1$-surgeries on these unknots on the pages will yield an open book decomposition of $S^1 \times S^2$ whose monodromy is the old monodromy $\phi = Id$ composed with a right-handed Dehn twist along each unknot. Moreover, by [5], this open book supports a Stein fillable and hence a tight contact structure on $S^1 \times S^2$. After the surgeries, the unknot $U_K$ on the page will be isotopic to the knot K and it can be Legendrian realized. The proof extends for any given link type. The page of the open book for a link can be constructed by taking the connected sum of the components of the unknotted link. $\square$

Remark 2.1

Note that if we follow the algorithm from the proof of Theorem 1.1 for $L(p, 1)$, then the surgery framing of U in Fig. 5 will be $-p$. To arrange the surgery framing of U to be 0, we need to blow up p times, by adding p unknots with framing $+1$. In this case, $+1$-framed unknots contribute negative Dehn twists to the monodromy of the open book yielding an overtwisted contact structure on $L(p, 1)$ (it is easy to find a sobering arc in this case, [6]). This is the reason why we cannot generalize Theorem 1.1 to $L(p, 1)$. See Fig. 6.

Fig. 6

Unknots with $+1$ framing contribute negative Dehn twists to the monodromy

Proof of Theorem 1.3

Let K be a knot in $L(p, 1)$ decomposed in terms of the standard generators of the pure braid group and let U be the unknot with surgery framing $-p$ decomposed as in Fig. 3. To construct an open book decomposition for $L(p, 1)$, we arrange the surgery framing of U to be $-1$. To this end, we blow up at the indicated points in Fig. 7, by adding $+1$-framed unknots at $2n-2$ points. This adds $+(2n-2)$ to the surgery framing $-p$.

Fig. 7

Arranging the framing of U

Next, we continue blowing up as in Fig. 8 to arrange the framing of any unknot having linking number 1 with K to be 0.

Fig. 8

Each unknot linking K has framing 0

Now, to arrange the surgery framing of U to be $-1$, we blow up by adding $p-2n+1$ unknots with framing $+1$ as in Fig. 9. This is possible only when $-p+(2n-2) < 0$. In addition, we arrange the framing of any unknot having linking number 1 with U to be 0 in Fig. 9 (if $-p+(2n-2) \ge 0$, then to arrange the framing of U to be $-1$, we need to add unknots with framing $-1$ and to continue to arrange these to have framing 0, we need to add unknots with framing $+1$. However, similar to the case of Remark 2.1, this yields an open book decomposition that supports an overtwisted contact structure).

Finally, we unknot the knot K by using the algorithm from [7, Remark 5.1]. We denote the unknotted K by $U_K$. Thus, we get a collection of links of unknots having framing 0 or framing $-1$ only, a Kirby diagram corresponding to the open book decomposition. A page of an open book decomposition is constructed by taking the connected sum of the disk bounded by $U_K$ and the disk bounded by U. We arrange each 0-framed unknot to puncture each disk page transversely once to form a binding component after the surgery. Each $-1$-framed unknot (in particular the unknot U) is arranged to be isotoped onto one of the punctured disk pages to perform the surgery on the page. Since each $-1$-framed unknot contributes positive Dehn twists to the monodromy, we get an open book decomposition for L(p, 1) supporting a tight contact structure. After the surgeries, $U_K$ will be isotopic to K and we can Legendrian realize K on the page. $\square$

Fig. 9

Each unknot linking U has framing 0 and U has surgery framing $-1$

We now give illustrative examples.

Example

Let K be a knot in L(4, 1) as in Fig. 10 where the knot K and the unknot U are decomposed in terms of the standard generators of the pure braid group.

Fig. 10

A knot K in L(4, 1). K is presented as a 4-pure braided plat and $K \cup U$ is decomposed in terms of the standard generators of the pure braid group

Follow the algorithm from [7, Theorem 5.2] for K and follow the algorithm from Theorem 1.3 for U and obtain Fig. 11 where the unknot U has the framing $-1$ at the end.

Fig. 11

Unknotting $K \cup U$

A page of an open book decomposition is constructed by taking the connected sum of the disk bounded by $U_K$ and the disk bounded by U. We can now isotope $-1$-framed unknots onto a page using the band for the connected sum. We obtain a planar open book decomposition for L(4, 1) where $U_K$ and each $-1$-framed unknot sit on the page and each 0-framed component forms a binding component as in Fig. 12. The page of the open decomposition is a disk with seven punctures. Note that after performing the surgeries, each $-1$-framed unknot contributes a right-handed Dehn twist to the monodromy of the open book decomposition and the unknot $U_K$ will be isotopic to the knot K. By [5], this open book supports a Stein fillable and hence a tight contact structure on L(4, 1). We can Legendrian realize each $-1$-framed unknot and K on the page of this open book.

Fig. 12

Taking the connected sum of the disks bounded by $U_K$ and U to construct a page

Example

Let K be a knot in $S^1 \times S^2$ where K and U are decomposed in terms of the standard generators of the pure braided group as in Fig. 13.

Fig. 13

A knot K in $S^1 \times S^2$. K is presented as a 6-pure braided plat and $K \cup U$ is decomposed in terms of the standard generators of the pure braid group

By following the proof of Theorem 1.1, unknot the knot K and call it $U_K$. Continue blowing up at the points indicated in Fig. 14 so that U has linking number 1 with $U_K$ and has framing 0 at the end. Further arrange all 0-framed unknots to have linking number 1 with $U_K$ as in Fig. 15 (and hence each 0-framed unknot punctures the disk bounded by $U_K$).

Fig. 14

Unknotting $K \cup U$

A page of an open book decomposition for $S^1 \times S^2$ containing K is constructed from the disk bounded by $U_K$ in $S^3$, see Fig. 15. The 3-sphere $S^3$ has a natural open book decomposition with this disk page and with the monodromy $\phi = Id$. After performing the 0-surgeries, each unknot with framing 0 (also U) forms a binding component. The page of the open book decomposition is a disk with nine punctures. We isotope all $-1$-framed unknots onto a page. After performing $-1$-surgeries, we will get an open book decomposition of $S^1 \times S^2$ whose monodromy is the old monodromy $\phi = Id$ composed with a right-handed Dehn twist along each $-1$-framed unknot. Pushing $U_K$ off of itself inside the page is a way to see $U_K$ on the page. After the surgeries, $U_K$ will be isotopic to K and K can be Legendrian realized on the page.

Fig. 15

Passing from a surgery diagram to an open book decomposition

3 Final remarks and questions

Remark 3.1

By [7, Lemma 5.4], any Legendrian knot in any tight lens space L(p, 1) or in the tight $S^1 \times S^2$ with a Thurston-Bennequin invariant $\mathtt{tb}(L) > 0$ has $sg(L) > 0$. Thus, the Legendrian representative constructed in Theorem 1.1 or in Theorem 1.3 is a Legendrian knot having the Thurston-Bennequin invariant less than or equal to zero. Thus, it would be very interesting to know the answers to the following questions.

Question 3.2

Let L be a Legendrian knot in tight $(L(p, 1), \xi )$ with $\mathtt{tb}(L) \le 0$. For all such Legendrian knots, is $sg(L) = 0$?

Question 3.3

Is there a contact 3-manifold where all Legendrian knots have $sg(L)=0$?

Question 3.4

Do all contact 3-manifolds contain a Legendrian knot L with $sg(L)>0$?

Question 3.5

Does there exist a contact 3-manifold containing a Legendrian knot with $sg(L)=2$? Construct explicit examples of Legendrian knots with $sg(L)=2$.

Question 3.6

Can a Legendrian knot have arbitrary large support genus?

There are many operations one can perform on knots: connected sum, cabling, Whitehead doubling, Legendrian satellite operation. Along these lines:

Question 3.7

Which operations on knots preserve the support genus? Is there an operation on knots which increases/decreases the support genus?

Question 3.8

Which contact surgeries preserves the support genus?

Another interesting question is

Question 3.9

For which knot types in a contact 3-manifold are all Legendrian knots with maximal Thurston-Bennequin invariant determined by their support genus invariant?

Remark 3.10

The planar open book constructed in Theorem 1.3 supports some tight contact structure on $L(p, 1)$ which we can not keep track of from the algorithm. It would be very interesting to find an algorithm that works for all possible tight contact structures on lens spaces and which puts the given knot on the page and keeps the open book decomposition planar. Note that since $S^1 \times S^2$ has a unique tight (and Stein fillable) contact structure, such a problem does not appear for $S^1 \times S^2$.

Question 3.11

Do all knot types in any tight $L(p, 1)$ have a Legendrian representative L such that $sg(L) = 0$?

It is not (a priori) clear that Theorem 1.3 extends to general lens spaces.

Question 3.12

For $p> q >0$, do all knot types in any tight L(p, q) have a Legendrian representative L such that $sg(L) = 0$?