The Self-Linking Number in Annulus and Pants Open Book Decompositions
We find a self-linking number formula for a given null-homologous transverse link in a contact manifold that is compatible with either an annulus or a pair of pants open book decomposition. It extends Bennequin’s self-linking formula for a braid in the standard contact -sphere.
Key words and phrases:
2000 Mathematics Subject Classification:Primary 57M25, 57M27; Secondary 57M50
Alexander’s theorem  states that every closed and oriented -manifold admits an open book decomposition.
Let be a surface with non empty boundary and be a diffeomorphism of the surface fixing the boundary pointwise. We construct a closed manifold
where “” is an equivalence relation satisfying for and for and . The pair is called an abstract open book decomposition of the manifold .
Alternatively, an open book decomposition for can be defined as a pair (L, ), where (1) is an oriented link in called the binding of the open book; (2) is a fibration whose fiber, , called a page, is the interior of a compact surface such that for all .
One of the central results about the topology of contact -manifolds is Giroux correspondence :
For example, the standard contact structure on corresponds to the open book decomposition .
We define a braid and the braid index in a general open book setting:
Suppose is an open book decomposition for a -manifold . A link is called a (closed) braid if transversely intersects each page of the open book. That is, at each point , we have . The braid index of a braid is the degree of the map restricted to . In other words, if a braid intersects each page in points, then the braid index of is .
Bennequin  proved that any transverse link in can be transversely isotoped to a closed braid in . Later the second author generalized Bennequin’s result into the following:
[13, Theorem 3.2.1] Suppose is an open book decomposition for a -manifold . Let be a compatible contact structure. Let be a transverse link in . Then can be transversely isotoped to a braid in .
The self linking Bennequin number is a classical invariant for transverse knots. Bennequin  gave a formula of the self linking number for a braid in :
where is the braid index, and the algebraic crossing number (the exponent sum) of the braid.
The first goal of this paper is to give a combinatorial description for the self linking number of a null-homologous transverse link in the contact lens spaces compatible with the annulus open book decomposition with monodromy the power of the positive Dehn twist . By Theorem 1.3, our problem is reduced to searching a self linking formula for a null-homologous braid in the open book decomposition . Such a braid is given by a product of permutations of points in a local disk on the annulus and moves of points which turn around the hole of . We denote by the algebraic crossing number of the local permutations, and by the algebraic rotation number around the hole of , see Definition 2.5 for precise definitions. With these notations, we extend Bennequin’s formula (1.1) into the following:
Theorem 4.1. Let be a null-homologous closed braid in of braid index . For we have
When there exists a canonical Seifert surface of and we have
The Seifert surface will be constructed in Section 3. The surface is canonical in the sense that the way of construction is similar to that of the standard Seifert surface, or Bennequin surface, of a closed braid in .
Our second goal is to find a self-linking formula for null-homologous transverse links in a contact Seifert fibered manifold of signature . Let be a pair of pants (a disk with two holes). Let () be the positive Dehn twists along the curves parallel to the boundary circles of . Then has an open book decomposition , and is equipped with a compatible contact structure. A braid in the pants open book is a product of permutations of points in a local disk on and moves of points which turn around the holes of . We denote by the algebraic crossing number of the local permutations and by () the algebraic winding number around the holes. See Definition 5.4 for precise definitions. We obtain the following formula which also extends (1.1).
Theorem 5.6 Let be a null-homologous braid in of braid index . Suppose are integers with ; ; or . We have:
where is some Seifert surface for . The constants , are determined by , , , and , under the assumption that is null-homologous, see Definition 5.4.
The organization of the paper is the following:
In Section 2, we fix notations and study properties of the contact lens space .
In Section 3, we construct a Bennequin type Seifert surface for a given braid in . In general, this is an immersed surface and the Bennequin-Eliashberg inequality is not satisfied even for tight cases. We resolve all the singularities and obtain an embedded surface . We develop a theory about resolution of singularities of an immersed surface and corresponding changes in characteristic foliations.
In Section 4, we prove Theorem 4.1, an explicit formula of the self linking number relative to , which extends Bennequin’s formula (1.1). As the self linking number is defined to be the euler number of the contact -plane bundle relative to the surface framing, we measure the difference between the immersed -framing and the embedded -framing. We also study the behavior of our self linking number under a braid stabilization. Corollary 4.5 states that our self linking number is invariant under a positive stabilization and changes by under a negative stabilization, which extends Bennequin’s result for braids in .
Acknowledgements. The authors would like to thank John Etnyre for numerous useful comments and sharing his ideas, especially those on Corollary 3.9, and Matthew Hedden for helpful comments on Section 4. They also thank the referee for carefully examining the paper and providing constructive comments. K.K. thanks Tim Cochran and Walter Neumann for stimulus conversations.
Let be an annulus and the positive Dehn twist about the core circle . For simplicity, we denote by .
We study an abstract open book decomposition .
The corresponding manifold to is:
Let be a disk and . Recall that is a planar open book decomposition for . Let be a disc with boundary . The core of the solid torus is the unknot, . The meridian of the torus is . Pick a point , and define a longitude of as . Remove from , and attach a new solid torus by identifying its meridian with and its longitude with . This is the -surgery along the unknot . The resulting manifold is . In this way we get an open book decomposition for , whose page is the union of the annulus , shaded in Figure 2-(1), and the annulus bounded by and the core of the solid torus, sketched in Figure 2-(2).
The Dehn twist about the core , sketched in Figure 2-(3), of the page annulus is equivalent to applying -surgery along the unknot . The link is the positive Hopf link. By the slam-dunk operation, the surgery description is reduced to the -surgery along , which represents when and when . ∎
Let be the contact manifold corresponds to the open book .
The contact manifold is overtwisted if and only if . When , this is the unique tight contact structure for .
If , Goodman’s criterion for overtwistedness [10, Theorem 1.2] implies that is overtwisted.
When , according to [6, proof of Lemma 3.2] of Etnyre-Honda, the open book is a boundary of a positive Lefschetz fibration on a -manifold , so that is Stein filled by , hence tight. Moreover, is the unique tight contact structure on due to Eliashberg .
When , the monodromy is a product of positive Dehn twists. Etnyre-Honda’s [6, Lemma 3.2] guarantees that the contact structure compatible with such an open book is Stein fillable, hence tight. The uniqueness for follows from Honda’s classification of tight contact structures for lens spaces . More precisely, we have
and , thus the manifold has the unique tight contact structure. ∎
We fix notations. See Figure 3. Suppose we have a null-homologous closed braid of braid index in the open book . Let whose orientations are induced by that of . Let () denote the page . Under the identification , we set . Let be a circle between and which is oriented clockwise.
Choose points sitting between and . By braid isotopy, which preserves the transverse knot class (Theorem 2.8-(2)), we may assume that:
Let () be the generators of Artin’s braid group satisfying and for . Geometrically, acts by switching the marked points and counterclockwise. The circle will appear in Section 3.1. Let be a braid element which moves once around the annulus in the indicated direction.
An -strand braid in has a braid word in .
Let be concentric disks of center . Identify the annulus with and . Consider the union , which we identify with an -strand braid in Artin’s braid group . Let be the projection onto the first factor. Up to homotopy, we can think that is a (non-simple) closed curve in . Denote its homotopy class by
Let be generators of as in Figure 4-(1).
The transition from Figure 4-(2) to (3) shows:
Since our is equal to in the braid group , the braid can be written in letters . Since , the statement of the proposition follows. ∎
Let (resp. ) be the exponent sum of ’s (resp. ) in the braid word of .
If (resp. ), we may assume that (resp. ).
To prove Proposition 2.6, we first define braid stabilization and recall its properties.
Let be a closed braid in an open book . Suppose that is one of the bindings of the open book and is a point, see Figure 5. Join and a point on by an arc . A positive negative stabilization of about along is pulling a small neighborhood of of the braid, then adding a positive (negative) kink about in a neighborhood of .
The second author proved Markov theorem in a general open book setting:
[13, Theorem 4.1.3 and 4.1.4]
Two closed braids and in an open book decomposition have the same topological type if and only if they are related by braid isotopy, positive and negative braid stabilizations.
The above are transversely isotopic if and only if they are related by braid isotopy and positive braid stabilizations.
Proof of Proposition 2.6.
Suppose is an -strand braid. Recall that has two binding components, and . Let be an arc joining and and intersecting at a point as sketched in Figure 6.
Pick a small line segment of the strand in , near the top page of the open book, and positively stabilize it along . As a consequence, it gains a new braid strand, which we call , lying in a small tubular neighborhood of , see Figure 7-(1).
Put a point on the right side of between and define a braid generator as in Figure 6. Move by a braid isotopy supported in so that intersects the page at . This isotopy introduces in as a consequence of the monodromy . Compare Figure 7-(1) and (2).
We observe that in a stabilized braid, plays the role of the old and as Figure 7-(3) shows, they are related by:
Thus a positive stabilization about takes a word to where is obtained from replacing each with . The data change in the following way:
Theorem 2.8-(2) tells that a positive stabilization preserves the transverse knot type, so if (resp. ) we may assume that (resp. ). ∎
The next corollary introduces a number :
If there exists a non-negative integer such that . If then .
3. Construction of Seifert surface
The goal of this section is to construct a Seifert surface for a null-homologous braid whose braid word is written in . (By abuse of notation, we use for both the closed braid and its braid word.) We first construct a surface and change it to . We further deform and finally obtain .
3.1. Construction of the surface
Let be a line segment perpendicular to having as one of its endpoints and with the other end on , see Figure 3. Since is disjoint from the Dehn twist curve , in the resulting manifold, , the arc swipes a disk . See Figure 8. The center of is . We orient so that the binding is positively transverse to .
Suppose the braid word for has length . If the () letter is (resp. ) then we join the disks and by a positively (resp. negatively) twisted band embedded in the set of pages . See Figure 9-(1).
If the letter is (resp. ), then we attach to the disk an annulus embedded in . We call such an annulus an -annulus. See Figure 9-(2). Let be an oriented circle between circles and as sketched in Figure 3. One of the boundaries of each -annulus represents (resp. ) and becomes part of the braid . The other boundary, which we denote by (resp. ), is in .
We call the resulting surface .
By [8, Proposition 4.6.11], we may assume that the characteristic foliation of our surface is of Morse-Smale type. Each disk has a positive elliptic point. A positive (negative) band between the -disks contributes one positive (negative) hyperbolic point. The foliation on the disk together with an attached -annulus has a positive (resp. negative) hyperbolic singularity as sketched in Figure 10-(1) (resp. (2)) if the corresponding braid word is (resp.
3.2. Construction of the surface
In Section 3.1, we have constructed an embedded oriented surface whose boundary consists of the braid and copies of ’s. Let (resp. ) be the exponent sum of ’s (resp. ’s) in the braid word for . Let be the number of ’s appearing in the braid word for of length (i.e., ). Then there exist and with such that
By attaching vertical annuli to pairs of and circles as described in Figure 11, we can construct an embedded oriented surface, which we call , whose boundary consists of
Suppose that and
If ( and do not coexist in the braid word for ), then take .
Else, let be the smallest index for which . We attach a “vertical” annulus to as sketched in Figure 11. The boundary of the newly obtained surface (call this surface ) contains two less -curves:
but it preserves the sum:
Renumber the the boundary components,
then repeat the procedure for . If is the smallest index for which , then attach the annulus by nesting it inside the one previously attached. See the right sketch in Figure 12.
3.3. Construction of the immersed surface
We have constructed a surface satisfying the boundary condition (3.1). In particular, when we have already obtained an embedded surface whose boundary is . Define .
When , we construct an immersed surface from , by attaching disks about the binding .
Assume that . Proposition 2.6 justifies assuming . Let be the -annuli whose boundaries contribute to the copies of -circles as in Proposition 3.1. Recall the number defined in Corollary 2.9. Let be arcs, see Figure 3, disjoint from the Dehn twist circle , such that one end of each sits on the binding . Let be disks, called -disks, obtained by swiping in the open book so that the center of is pierced by . For each , connect smoothly with annuli by copies of the twisted band as in Figure 13-(1). When , attach twisted bands as in Figure 13-(2). We have obtained an immersed surface, which we denote by , see Figure 14.
Regardless of the sign of , all the singularities for the characteristic foliation of the attached bands and the -disks are positive elliptic.
The surface has additional positive elliptic points compared to .
By definition of -disk, its characteristic foliation has a single singularity at its center and it is of elliptic type (Figure 13). The orientation of -disk is induced from that of the -annuli so that the sign of the elliptic point is positive regardless of the sign of .
In the following, we show that there are no hyperbolic points on the twisted band. See Figure 15. Parameterize the twisted band as . Attach the side of the band to the -disk and to the -circle so that the (dashed) line segment sit on one page of the open book. We make the resulting surface smooth near the two attaching sides of the band.
Take points on the twisted band , , and . Let be a tangent vector (red arrow) of the band at that is perpendicular to the dashed line .
When , with respect to the page of the open book, is vertical, is slanted , and is slanted -degree ( because the braid transversely intersects all the pages of the open book positively. Next we look at contact planes (light blue line segment) at . In Figure 15, the positive side of a contact plane is marked “”. At each point of the bindings , we may assume that the contact plane is positively perpendicular to the binding. Between the bindings, the contact planes rotate counter clockwise along the radial lines. Since is close to the binding , for some , is slanted -degree with respect to the page of the open book. While, is slanted -degree for some since is on the circle which is between and (Figure 3). At any point between and on the dashed line, the tangent plane is slanted more than the contact plane. It means that they never coincide. Since the band is a small neighborhood of the dashed line, contact planes are never tangent to the band, hence there are no singularities in the characteristic foliation on the twisted band.
When , at , the tangent vector is slanted -degree and the contact plane is slanted -degree. Braid is a transverse link so it intersects contact planes positively. Considering that is close to the braid (orange arc), intersects positively, i.e., . Therefore, the tangent planes and the contact planes never coincide along the dashed line from to , hence there are no singularities in the characteristic foliation on the twisted band. ∎
3.4. Construction of the immersed surface
Let be a closed braid in of braid index . The immersed surface constructed in Section 3.3 has boundary in . Each closed curve representing bounds a disk about the binding . We call it a -disk, see Figure 16.
There, the spirals in the bottom annulus page are identified, via the Dehn twist , with the straight line segments in the top annulus page. There are -copies of -disk and they are disjoint from each other. Since -disks are nearly ‘vertical’ as in Figure 16, the tangent planes and the contact planes, which rotate counter clockwise along the radial lines from to , intersect transversely. This means that the characteristic foliation of each -disk has a single singularity, which occurs at the intersection point with and whose type is elliptic. The orientations of the -disks are compatible with those of the boundaries so that the -disks and intersect negatively. Therefore the signs of the elliptic points are negative.
We construct an immersed surface by glueing the -disks and along the copies of the curve.
This has additional negative elliptic singularities given by the -disks compared to the surface .
3.5. Resolution of singularities
We start this section by defining three types of intersection of surfaces; branch, clasp, and ribbon, then study resolution of self-intersections.
Let be an immersed oriented surfaces with given by the immersion . Let be a simple arc where intersects itself, and denote by and the endpoints of .
If is sitting on , and is a branch point of a neighborhood Riemann surface, see Figure 17-(1), then we call a branch intersection.
Next assume that the preimage of , , consists of two arcs, say . Denote the end points of by and for .
If and then we call the intersection a clasp intersection. A local picture of is the left sketch of Figure 18.
If and then we call the intersection a ribbon intersection. See the right sketch of Figure 18.
See Figure 19. The immersed surface, , has:
branches formed by -annuli and -disks.
clasp intersections when . Recall that is obtained by attaching copies of -disk about the binding . Each pair among these disks interacts as in Figure 19 giving rise to clasp intersections. When , there are no clasps.
several ribbon intersections of -disks and the (nested) vertical annuli of Figure 11.
In Section 3.6, we resolve these self-intersections to obtain an embedded surface .
In the following, we assume that is a transverse knot in a contact manifold and an immersed oriented surface with . Also, we assume that (i) the self-intersection set of consists of ribbon, clasp, or branch intersections; (ii) the characteristic foliation is of Morse-Smale type.
Let be a self-intersection arc. Near a point , intersects itself transversely as in Figure 20-(1). Let () be surfaces meeting at . The orientation of is induced from that of . Resolve the singularity by cutting out along and re-gluing along and along so that the orientations of the surfaces agree. See Figure 20-(2). Call the new surface .
We orient the leaves of the characteristic foliation following [12, page 80]: For a nonsingular point of a leaf of the foliation, let be a positive normal vector to . We choose a vector so that is a positive basis for . This vector field determines the orientation of the characteristic foliation.
We observe that if both and transversely intersect the line (Figure 20-(1)), then the orientations of and agree at . Hence, after the cut and glue operation, the new characteristic foliation is obtained by smoothly connecting the old and , and also and . See Figure 20-(2).
The signs of the new hyperbolic points are determined in the following way:
Suppose that and both and