Rectangular diagrams of surfaces: representability

Rectangular diagrams of surfaces: representability

Ivan Dynnikov and Maxim Prasolov Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina Str., Moscow 119991, Russia dynnikov@mech.math.msu.su 0x00002a@gmail.com
Abstract.

We introduce a simple combinatorial way, which we call a rectangular diagram of a surface, to represent a surface in the three-sphere. It has a particularly nice relation to the standard contact structure on and to rectangular diagrams of links. By using rectangular diagrams of surfaces we are going, in particular, to develop a method to distinguish Legendrian knots. This requires a lot of technical work of which the present paper addresses only the first basic question: which isotopy classes of surfaces can be represented by a rectangular diagram. Vaguely speaking the answer is this: there is no restriction on the isotopy class of the surface, but there is a restriction on the rectangular diagram of the boundary link that can arise from the presentation of the surface. The result extends to Giroux’s convex surfaces for which this restriction on the boundary has a natural meaning. In a subsequent paper we are going to consider transformations of rectangular diagrams of surfaces and to study their properties. By using the formalism of rectangular diagrams of surfaces we also produce here an annulus in that we expect to be a counterexample to the following conjecture: if two Legendrian knots cobound an annulus and have zero Thurston–Bennequin numbers relative to this annulus, then they are Legendrian isotopic.

The work is supported by the Russian Science Foundation under grant 14-50-00005 and performed in Steklov Mathematical Institute of Russian Academy of Sciences.

1. Introduction

Rectangular diagrams of links, also known as arc-presentations and grid diagrams, have proved to be useful tool in knot theory. It is shown in [5] that any rectangular diagram of the unknot admits a monotonic simplification to a square by elementary moves. This was used by M. Lackenby to prove a polynomial bound on the number of Reidemeister moves needed to untangle a planar diagram of the unknot [22].

It is tempting to extend the monotonic simplification approach to general knots and links, but certainly it cannot be done straightforwardly. Typically a non-trivial link type admits more than one rectangular diagram that cannot be simplified further by elementary moves without using stabilizations. As shown in [6] classifying such diagrams is closely related to classifying Legendrian links not admitting a destabilization.

In particular, the main result of [6] implies that the finiteness of non-destabilizable Legendrian types in each topological link type (which is the matter of Question 61 in [4]) is equivalent to the finiteness of the number of rectangular diagrams that represent each given link type and cannot be monotonically simplified. Neither of these has been established so far.

Surfaces embedded in , either closed or bounded by a link, have always been one of the key instruments of the knot theory. For studying contact structures and Legendrian links it is useful to consider surfaces that are in special position with respect to the contact structure, so called convex surfaces. They were introduced by E. Giroux in [13] and studied also in [3, 10, 18, 21, 24].

Sometimes one can prove the existence of a convex surface having certain combinatorial and algebraic properties for one contact structure and non-existence of such a surface for the other, thus showing there is no contactomorphism between the structures. This method can be viewed as a generalization of the classical work of Bennequin [1] where he distinguishes a contact structure in from the standard one by showing that an overtwisted disc exists for the former and does not exist for the latter. Indeed, an overtwisted disc can be viewed as a disc with Legendrian boundary and a closed dividing curve.

This method can also be used to distinguish Legendrian links by applying the argument to the link complement. For instance, some Legendrian types of iterated torus knots are distinguished in [11] by showing that certain slopes on an incompressible torus in the knot complement are realized by dividing curves for one knot and not realized for the other.

Note that whereas the problem of algorithmic comparing topological types of two links has been solved (see [26]) no algorithm is known to compare Legendrian types of two Legendrian links having the same topological type. Examples of pairs of Legendrian knots for which the equivalence remains an open question start from six (!) crossings [2].

In this paper we propose a simple combinatorial way to represent compact surfaces in so that the boundary is represented in the rectangular way. The basic idea of this presentation is not quite new and is implicitly present in the literature. In particular, it can be considered as an instance of Kneser–Haken’s normal surfaces for triangulations of obtained by the join construction from triangulations of two circles .

We observe in this paper that the discussed combinatorial approach appears to be well adapted to Giroux’s convex surfaces with Legendrian boundary. In particular, we show that any convex surface with Legendrian boundary is equivalent (in a certain natural sense) to a one that can be presented in the proposed way.

By using this approach we are going to distinguish some pairs of Legendrian knots that are not distinguishable by known methods. Since a rectangular diagram of a surface is a simple combinatorial object, sometimes the non-existence of a diagram with required properties can be established by combinatorial methods. Combined with the representability result of the present paper this may be used to show the non-existence of a convex surface with certain properties in the complement of a Legendrian knot. If a convex surfaces with the same properties is known to exist for another Legendrian knot the two Legendrian types must be different. This method is out of the scope of the present paper and will be presented in a subsequent paper.

We also consider here in more detail embedded convex (in Giroux’s sense) annuli in tangent to the contact structure along the boundary. Such annuli are the main building blocks of closed convex surfaces in . We discuss the following specific question: are the two boundary components of such an annulus always equivalent as Legendrian knots?

This question appears to be highly non-trivial. An attempt to answer it in the positive was made in manuscript [17] but a complete proof was not given. We believe that the answer is actually negative, but our attempt to find a simple counterexample has failed. In a number of examples that we tried the two boundary components of the annulus appeared to be Legendrian equivalent, and a Legendrian isotopy was easily found. Namely, the boundary components were transformed to each other by elementary moves preserving the complexity of the diagram (exchange moves).

We propose here an example of an annulus of the above mentioned type whose boundary components cannot be transformed to each other without using stabilizations. We conjecture that the two Legendrian knot types in our example are different.

Remark 1.

Another use of rectangular diagrams of links comes from their relation to Floer homologies. As C. Manolescu, P. Ozsváth, and S. Sarkar show in [23] the definition of the knot Floer homology for links presented in the grid form can be given in purely combinatorial terms. Up to this writing, we are not aware of any useful interaction between rectangular diagrams of surfaces and algebraic theories. It would be interesting to see some.

The paper is organized as follows. In Section 2 we introduce the main subject of the paper, rectangular diagrams of surfaces, and other basic related objects. We show that every isotopy class of a surface can be presented by a rectangular diagram, but rectangular presentations of the boundary that will arise in this way satisfy certain restrictions. In Section 3 we discuss Legendrian links and graphs and their presentations by rectangular diagrams. They are used in the formulation and the proof of our main result—on the representability of convex surfaces by rectangular diagrams—which occupy Section 4. At the end of Section 4 we discuss the above mentioned conjecture about annuli with Legendrian boundary.

Acknowledgement

We are indebted to our anonymous referee for very careful reading of our paper, which has led to many clarifications in the text.

2. Rectangular diagrams of a surface

2.1. Definitions

For any two distinct points , , say, of the oriented circle we denote by the closed arc of with endpoints , such that if it is endowed with the orientation inherited from , see Fig. 1.

Figure 1. Intervals and on the oriented circle

By we denote the corresponding open arc: .

Definition 1.

A rectangle in the -torus is a subset of the form , where , , .

Two rectangles , are said to be compatible if their intersection satisfies one of the following:

  1. is empty;

  2. is a subset of vertices of ;

  3. is a rectangle disjoint from the vertices of both rectangles and .

A rectangular diagram of a surface is a collection of pairwise compatible rectangles in  such that every meridian and every longitude of the torus contains at most two free vertices, where by a free vertex we mean a point that is a vertex of exactly one rectangle from .

To represent such a diagram graphically, we draw as a square with identified opposite sides, so, some rectangles are cut into two or four pieces,

Figure 2. A rectangle in

see Fig. 2.

One can see that for any pair of compatible rectangles one of the following three mutually exclusive cases occurs:

  1. the rectangles are disjoint;

  2. the rectangles share , , or vertices and are otherwise disjoint;

  3. the rectangles have the form (possibly after exchanging them) , with

Figure 3. Compatible rectangles

In the latter case we draw as if it passes over , see Fig. 3.

An example of a rectangular diagram of a surface is shown in Fig. 4 on the left. The free vertices are marked by small circles.

Figure 4. A rectangular diagram of a surface and its boundary
Definition 2.

By a rectangular diagram of a link we mean a finite set of points in such that every meridian and every longitude of contains no or exactly two points from . The points in are referred to as vertices of .

When is presented graphically in a square, the vertical and horizontal straight line segments connecting two vertices of will be called the edges of . Formally, an edge of is a pair of vertices lying on the same longitude or meridian of . Such vertices are said to be connected by an edge, and this will have literal meaning in the pictures.

A rectangular diagram of a link is said to be connected or to be a rectangular diagram of a knot if the vertices of can be ordered so that any two consecutive vertices are connected by an edge.

A connected component of a rectangular diagram of a link is a non-empty subset that is a rectangular diagram of a knot.

Definition 3.

Let be a rectangular diagram of a surface. The set of free vertices of will be called the boundary of and denoted by .

It is readily seen that the boundary of a rectangular diagram of a surface is always a rectangular diagram of a link. In particular, for any rectangle the boundary of the diagram is the set of vertices of . It should not be confused with the boundary of the rectangle itself, which is understood in the conventional sense.

Fig. 4 shows a rectangular diagram of a surface (left) and its boundary (right) with edges added. The reason for drawing some edges passing over the others will be explained below.

2.2. Cusps and pizza slices

Here we introduce the class of topological objects (curves and surfaces) we will mostly deal with.

Definition 4.

Let be a piecewise smooth simple arc or simple closed curve in and a point distinct from the endpoints. We say that has a cusp at the point if admits a local parametrization such that

The curve is called cusp-free if it has no cusps, and cusped if all singularities of are cusps.

The links in that we consider will always be cusp-free.

Definition 5.

We say that is a surface with corners if is a subset of a -dimensional submanifold with (possibly empty) boundary such that:

  1. the embedding is regular and of smoothness class ;

  2. is bounded in by a (possibly empty) collection of mutually disjoint piecewise smooth cusp-free simple closed curves.

This definition simply means that surfaces we want to consider may have corners at the boundary, but in a broader sense than one usually means by saying ‘a manifold with corners’. Namely, the angle at a corner can be arbitrary between and (exclusive), not necessarily between and see Fig. 5.

Figure 5. Arbitrary angles from are allowed at corners

However, surfaces with corners are not allowed to spiral around a point at the boundary as shown in Fig. 6.

Figure 6. Spiralling around a boundary point is forbidden for a surface with corners

For future use we also need the following definition.

Definition 6.

By a pizza slice centered at we mean a disc that has the form , where  is a surface with corners such that , and is a closed ball centered at with small enough radius such that has no singularities in and consists of a single arc. (The point may or may not be a singularity of .)

For such a pizza slice we also say that it is attached to the arc .

Two pizza slices , centered at the same point are said to be equivalent if for small enough the pizza slices , , are attached to the same arc and for we have , where denotes the distance function.

2.3. 3D realizations of rectangular diagrams

With every rectangular diagram of a link or a surface we associate an actual link or a surface, respectively, in as we now describe.

We represent as the join of two circles:

(1)

We also identify it with the unit sphere in as follows:

The triple will be used as a coordinate system in . The two circles defined by and will be denoted by and , respectively.

Definition 7.

By the torus projection of a subset we mean the image of under the map defined by .

For a point in we denote by the image of the arc in .

Definition 8.

Let be a rectangular diagram of a link. By the link associated with we mean the following union of arcs: .

One can see that this union is indeed a collection of pairwise disjoint simple closed piecewise smooth curves. Moreover, is a union of piecewise geodesic closed curves, which are cusp-free. One can also see that if is a connected component of , then is a connected component of  and vice versa.

To get a conventional, planar picture of a link isotopic to one cuts into a square and then join the vertices of by edges, letting the vertical edges pass over the horizontal ones, see the right picture in Fig. 4. Finally, we remark that is the torus projection of , and is the only link satisfying this property.

Cooking a surface out of a rectangular diagram of a surface is less visual, but the main principle is similar: the surface associated with a rectangular diagram of a surface should have the torus projection prescribed by the diagram. The surface associated with a rectangular diagram will be composed of discs associated with individual rectangles.

For a rectangle , the image of in under identifications (1) is the tetrahedron , which we denote by . We want to define as a disc in with boundary so that the union of such discs over all rectangles of a rectangular diagram of a surface yield a smoothly embedded surface. This can be done in numerous ways among which we choose one particularly convenient as it allows us to write down an explicit parametrization of  and behaves nicely in the respects addressed in Section 4.

We denote by a bounded harmonic function on the interior  of  that tends to as tends to or  while stays fixed, and tends to  as tends to or while stays fixed. Such a function exists and is unique, which follows from the Poisson integral formula and the uniformization theorem.

Remark 2.

The function admits an explicit presentation in terms of the Weierstrass elliptic function with half-periods and :

where , and are assumed to be reals satisfying , .

Definition 9.

We call the image in under identifications (1) of the closure of the following open disc in :

(2)

the tile associated with and denote it by .

Let be a rectangular diagram of a surface. We define the surface associated with to be the union  of the tiles associated with rectangles from : .

Proposition 1.

Let be a rectangular diagram of a surface. Then is a surface with corners in , and we have .

Proof.

First consider a single rectangle . The function can be extended continuously and smoothly to except at the vertices of , where it jumps by . The closure of the graph of is a smooth image of an octagon. Its boundary consists of four straight line segments parallel to the sides of and another four straight line segments hanging over the vertices of , see Fig. 7.

Figure 7. The graphs and of the functions and , respectively

Along each of the latter four straight line segments, is tangent to a helicoid that is independent of the size of (i.e. of and ): the tangent plane to at the point , , , is . Tangency with a helicoid is a general property of the graphs of bounded harmonic functions on a polygon that are constant on each side of the polygon and have jumps at corners. Near each corner such a function has a form , where and is a complex coordinate in the plane. The tangency with a helicoid can even be shown to be of the second order.

By composing with the map , which is a monotonic function such that and , we get the function , whose graph  is also a curved octagon with the same boundary as , but now it has an additional property that the tangent plane to  is orthogonal to the -plane along the whole of the boundary . The specific choice of will play an important role in the sequel.

Identification (1) takes to a disc with corners in , which is the tile . Fig. 8 shows a tile projected stereographically into . (The reader is alerted that the ‘coordinate grids’ in Fig. 7, which are added to visualize the surfaces, are not related to that in Fig. 8.) The boundary has exactly two points at each of the circles and , and at these points the tangent plane to is orthogonal to the corresponding circle.

Figure 8. A tile

These points will be referred to as the vertices of , and the four arcs of the form , where , which form the boundary of , the sides of .

By construction we also have the following.

Lemma 1.

The tile is tangent to the plane field

(3)

along the sides and , and to the plane field

(4)

along the sides and .

Let and be two compatible rectangles. If they are disjoint, then either the tiles and are disjoint or they share one or two vertices. In the latter case they have the same tangent plane at the common vertices.

If and share a vertex , say, then and share a side, which is , and all other tiles are disjoint from . The tiles and approach from opposite sides and have the same tangent plane at each point of . Moreover, the intersection has the form , where consists of common vertices of and (one, two, or four of them). Thus, the direction of the tangent plane to has no discontinuity at .

Figure 9 shows the preimages in of two tiles sharing a single side, and the tiles themselves.

Figure 9. Two tiles sharing a side

Finally, if is a rectangle, then the tiles and are disjoint. Indeed, by construction, at the horizontal sides of one of the functions , vanishes whereas the other is strictly positive, and at the vertical sides of one of these functions equals and the other is strictly less than . Thus, we have for all , and, therefore, the inequality also holds for all by the maximum principle for harmonic functions. Fig. 10 shows the relative position of the graphs of the functions and and the associated tiles in this case.

Figure 10. Two tiles corresponding to overlapping rectangles

Thus, in general, intersection of any two tiles , of bounds to the intersection of their boundaries, and the direction of the tangent plane to is continuous on the whole of . It remains to examine the boundary of and to verify that no singularities other than corners occur at .

The boundary of each tile consists of four arcs of the form , where is a vertex of . If is a vertex of two rectangles from , then two tiles of are attached to , so, the interior of is disjoint from . Thus, consists of all the arcs such that is a free vertex of . This implies .

Let be a point in . It corresponds to a longitude of the torus . Let this longitude be and denote it by . The tiles having as a vertex are associated with rectangles of the form or . Let be all such rectangles.

For small enough , the intersections are all pizza slices, and the only our concern is what their union looks like.

Since the rectangles are pairwise compatible, the arcs have pairwise disjoint interiors. The last condition in the definition of a rectangular diagram of a surface guarantees that is either the whole of or a single arc of the form .

In the former case, the union is ‘the whole pizza’, i.e. a disc contained in the interior of . In the latter case, this union is a pizza slice with the angle , , at .

The intersection of with the circle is considered similarly. ∎

Remark 3.

Recovering a surface from a rectangular diagram of a surface as described above may look somewhat counterintuitive. Indeed, if is a rectangle, the vertices of the tile correspond to the sides of and the sides of  to the vertices of .

However, as we will see, presentation of surfaces by rectangular diagram is quite practical. In particular, the class of all surfaces of the form is such that each surface in it is uniquely recovered from its torus projection, whose closure is the union .

2.4. Orientations

Quite often one needs to endow surfaces and links with an orientation. Here is how to do this in the language of rectangular diagrams.

Definition 10.

By an oriented rectangular diagram of a link we mean a pair in which is a rectangular diagram of a link and is an assignment of ‘’ or ‘’ to every vertex so that the endpoints of every edge are assigned different signs. The vertices with ‘’ assigned are referred to as positive and those with ‘’ assigned negative. We also say that is an orientation of .

Every orientation of a rectangular diagram of a link defines an orientation of the link by demanding that increases on the oriented arc whenever is a positive vertex of and decreases otherwise. One can readily see that this gives a one-to-one correspondence between orientations of a rectangular diagram of a link and those of the link associated with it.

Remark 4.

From the combinatorial point of view oriented rectangular diagrams of links is the same thing as grid diagrams, with X’s in the latter corresponding to positive vertices and O’s to negative ones.

Now we introduce orientations for rectangular diagrams of surfaces.

The pair of functions is a local coordinate system in the interior of each tile. If two tiles , , say, share a side, then the orientations defined by this system in and disagree at , see Fig. 9. So, it is natural to specify an orientation of a rectangular diagram of a surface as follows.

Definition 11.

An oriented rectangular diagram of a surface is a pair in which is a rectangular diagram of a surface and is an assignment ‘’ or ‘’ to every rectangle in so that the signs assigned to any two rectangles sharing a vertex are different. The rectangles with ‘’ assigned are then called positive and the others negative. The assignment is referred to as an orientation of .

Like in the case of links, orientations of any rectangular diagram of a surface are put in one-to-one correspondence with orientations of the surface by demanding being a positively oriented coordinate pair in the tiles corresponding to positive rectangles, and negatively oriented coordinate pair in the tiles corresponding to negative rectangles. In particular, does not admit an orientation if and only if is a non-orientable surface.

In what follows, to simplify the notation, we omit an explicit reference to orientations.

2.5. Framings

We will need a slightly more general notion of a framing of a link than the one typically uses in knot theory.

Definition 12.

Let be a cusp-free piecewise smooth link in . By a framing of we mean an isotopy class (relative to ) of surfaces with corners such that

  1. is a union of pairwise disjoint annuli ;

  2. for each , one of the connected components of is a component of and the other is smooth and disjoint from .

If is a framing of a link and is a collection of annuli representing , then we denote by the link . Roughly speaking, is obtained from by shifting along the framing . If is oriented, then is assumed to be oriented coherently with .

We also denote by the linking number , which is clearly an invariant of . If is a smooth knot then framings of are classified by (which is independent on the orientation of the knot).

But if has singularities, a framing contains more information. Indeed, at a singularity of , the tangent plane to any surface with corners such that is prescribed by . So, it cannot be changed by an isotopy of within the class of surfaces with corners.

The surface can approach such a singularity from two sides. Formally, this means that if is a singularity of and is an arc of the form with small enough , then there are two equivalence classes of pizza slices attached to , and the one that the surface realizes is an invariant of the framing.

Let a framing of be fixed. If is another framing of , then the twist of relative to  along any arc connecting two singularities of is a multiple of and is also a topological invariant of .

Clearly, these invariants determine the framing, and there are some obvious restrictions on them, which we need not to discuss.

If is a sublink, then the restriction of a framing is defined in the obvious way. Clearly, any framing of is defined uniquely by its restriction to every connected component of .

Definition 13.

If is a surface with corners we call the framing of defined by a collar neighborhood of the boundary framing induced by .

If is a sublink of , where is a surface then we denote by any link contained in such that bounds a collar neighborhood of in .

Definition 14.

Let be a rectangular diagram of a link. By a framing of we mean an ordering in each pair of vertices of forming an edge of .

For specifying a framing of a rectangular diagram of a link in a picture we proceed as follows. For every edge , we draw the arc if in the given framing and the arc if . Here by we mean if is a horizontal edge and if is a vertical one. The arcs are assumed to be drawn on , so in the actual planar picture some of these arcs get cut into two pieces, see Fig. 11.

Figure 11. Specifying a framing in the picture of a rectangular diagram of a link

As always, we draw vertical arcs over horizontal ones. The right picture in Fig. 12 shows an example of a framed rectangular diagram of a link.

Figure 12. A rectangular diagram of a surface and its boundary with boundary framing

By definition, any rectangular diagram of a link supports only finitely many framings. They encode framings of the corresponding link in that are not ‘twisted too much’. We now define them formally.

Definition 15.

Let be a rectangular diagram of a link. A framing of is called admissible if it can be presented by a surface tangent to one of the plane fields  defined by (3) and (4) at every point .

With every admissible framing of we associate (in general, non-uniquely) a framing of as follows.

Let be a union of annuli that represents an admissible framing of such that at every point the surface is tangent to or . Let be a point from . The tangent plane to at  is orthogonal to the corresponding since so are .

The intersection of with a small ball is a pizza slice, which is equivalent to one of the form or (depending on whether or ), where is the edge of corresponding to , and the equivalence  is given by (1). We denote the pizza slice  by . The framing of associated with is defined on the edge as .

Proposition 2.

For a generic , the construction above defines a one-to-one correspondence between admissible framings of and framings of . In general, this correspondence is one-to-many.

Proof.

Clearly, we can recover the equivalence classes of the pizza slices from the corresponding framing of , since the only information carried by the pizza slice is from which side it approaches the link . Namely, let and let be the corresponding horizontal edge. Then  is equivalent either to or to , and a choice of one of this options is precisely the information recorded in the framing of . Similarly for vertical edges.

The point now is that, for an admissible framing , the equivalence classes of ’s define completely. Indeed, let be a vertex of and , the endpoints of . At , , the tangent plane to is orthogonal to , and the framing of associated with prescribes from which side the surface approaches .

When traverses the planes and rotate around by and , respectively. Which way the tangent plane to must rotate is determined by and , see Fig. 13. So, the framing  is recovered uniquely.

Figure 13. Recovering an admissible framing from the pizza slices

The only possible reason for the correspondence between framings of and admissible framings of to not be a bijection is a situation when not all points in are singularities of . This occurs when some edges of have ‘length’ , which generically does not happen.

If this does happen, then the tangent plane to a surface representing can rotate freely around at any point that is not a singularity of , so when such a surface is required to be tangent to this does not necessarily prescribe from which side it should approach near . ∎

In the sequel we will assume that all rectangular diagrams of links that we consider are generic, and make no distinction between a framing of a diagram and an admissible framing of the corresponding link.

An obvious but important thing to note here is the following

Approximation Principle.

If is a framed rectangular diagram of a link and is the picture of  obtained as described right after Definition 14, then by smoothing near the vertices of we obtain the torus projection of a link of the form , in which the indication of passing over and under at the crossings corresponds to the relative position of the arcs in the -direction: the one with greater is overcrossing, see Fig. 14.

Figure 14. A framed rectangular diagram and the torus projection of

The principle also works in the other way. If is a link disjoint from the circles , then a rectangular diagram of a link such that is isotopic to can be obtained by approximating the torus projection of by the picture of a framed rectangular diagram of a link.

If is a rectangular diagram of a surface, then is tangent to or at every point of the boundary. This together with Approximation Principle implies the following.

Proposition 3.

Let be a rectangular diagram of a surface and . The boundary framing on induced by is admissible.

The picture of the corresponding framing of is obtained by connecting each pair of vertices of forming an edge by an arc in covered by the boundaries of the rectangles from , see Fig. 12.

2.6. Thurston–Bennequin numbers

Thurston–Bennequin number is a classical invariant of Legendrian links [1]. There are two Legendrian links associated with every rectangular diagram (see [28, 6] and also Section 3 below111The fact that arc-presentations ‘are Legendrian’ was mentioned by W. Menasco to the first present author already in 2003 and has been popularized since then), and their Thurston–Bennequin numbers can be computed from without any reference to the associated Legendrian links as we now describe.

Let be a(n oriented) rectangular diagram of a link. By we denote a diagram obtained from by a small shift in the -direction, i.e. a diagram of the form

where is so small that, whenever is a vertex of , there are no vertices of in the following four regions:

If is oriented, than inherits the orientation from .

Similarly we define by using a shift in the -direction.

Definition 16.

Let be an oriented diagram of a link. By the Thurston–Bennequin numbers of we mean the following two linking numbers:

These numbers do not change if the orientation of is reversed. So, if is a knot, then the numbers do not depend on the orientation of , and we can speak about Thurston–Bennequin numbers of even if is not oriented.

Proposition 4.

(i) Let be an oriented rectangular diagram of a link and a surface with corners in  such that . If is tangent to the plane field defined by (4) (respectively, defined by (3)) at all points of , then is equal to (respectively, to ).

(ii) The following equality holds

(5)

where denotes the number of vertices in .

(iii) The set

coincides with .

Proof.

Claim (i) follows from the fact that (respectively, ) is obtained from by a small shift in a direction transverse to (respectively, to ).

When traverses an arc of the form , where , the plane  rotates relative to the plane  by the angle . This implies Claim (ii).

We also see, that is maximized (respectively, minimized) over all admissible framings of if it can be presented by a surface with corners such that and is tangent to (respectively, to ) along . Thus for any admissible framing .

Let and . Define a framing as follows. For each horizontal edge of  with the negative vertex and the positive one, we put . Do the same for arbitrarily chosen vertical edges. For the remaining vertical edges put the opposite: .

Let be a surface representing the corresponding admissible framing of . There will be arcs of the form , , along which is tangent to and such arcs along which is tangent to . This implies . So, any integer in the interval can be realized by for a framing of , which completes the proof of Claim (iii). ∎

2.7. Which isotopy classes of surfaces can be presented by rectangular diagrams?

The following is a combinatorial definition of the relative Thurston–Bennequin number. The latter appears, e.g. in [19].

Definition 17.

Let be an oriented rectangular diagram of a link and a surface (which can be more general than a surface with corners) whose boundary contains . By the Thurston–Bennequin numbers of relative to we call

Again, if has a single connected component, then the numbers are independent of the orientation of .

Theorem 1.

Let be a rectangular diagram of a link and let be a compact surface in such that each connected component of is either disjoint from or contained in . Then a rectangular diagram of a surface  such that is isotopic to relative to exists if and only any connected component such that , has non-positive Thurston–Bennequin numbers relative to :

(6)

In this theorem is not necessarily assumed to be a surface with corners in the sense of Definition 5, and is allowed to have more general singularities at the boundary, both initially and during the isotopy. This allows to rotate it freely around , even near the singularities of .

Proof.

First, we show that condition (6) is necessary. Let be a rectangular diagram of a surface and  a connected component of . By Proposition 3 the boundary framing of  induced by  is admissible.

Therefore, by Proposition 4 for any connected component we have

which is equivalent to (6).

Now assume that (6) holds for any component of . We will use Lemma 2 and Proposition 5 proven below.

The link can have connected components disjoint from . By Lemma 2 we can isotop keeping  fixed so that will have the form for some rectangular diagram of a link . Moreover, we can ensure that the restriction of the boundary framing induced by to any connected component is admissible by making and smaller than and , respectively.

Thus, we may assume without loss of generality that contains (we achieve this by replacing with ). We can now isotop by altering it only in a small neighborhood of so that will become tangent either to or to at every point of .

Application of Proposition 5 with and completes the proof. ∎

Lemma 2.

Let be a rectangular diagram of a link, a link disjoint from , and an integer. Then there exists a rectangular diagram of a link such that

  1. the link is isotopic to relative to ;

  2. for any connected component of we have and .

Proof.

To show the existence of without the restrictions on the Thurston–Bennequin numbers we follow the Approximation Principle with a slight modification that now we should avoid intersections with the given link already given in the ‘rectangular’ form.

First, we perturb slightly to make it disjoint from both circles . We may also ensure that the torus projection of has only double transverse self-intersections.

The torus projection of the link is . The torus projection is disjoint from this set, since is disjoint from . Let be the distance between and .

We can now approximate by a framed rectangular diagram of a link in the -neighborhood of . By a small perturbation if necessary we can achieve that the edges of do not appear on the same longitudes or meridians as the edges of . The link will be isotopic to relative to , see Fig. 15.

Figure 15. Approximating a torus projection of a link in the complement of by a framed rectangular diagram of a link

(The framing that comes with plays no role in the sequel and should be discarded.)

Now if some of the Thurston–Bennequin numbers of connected components of are too large, we apply stabilizations to . Recall (see [5, 6]) that a stabilization of a rectangular diagram is a replacement of a vertex , say, by three vertices such that

  1. are vertices of a square of the form ;

  2. the square is so small that there are no other vertices of the diagram inside the regions and and there are exactly two vertices at the boundary of each.

We distinguish between type I and type II stabilizations as shown in Fig. 16.

Figure 16. Types of stabilizations and destabilizations

Type I stabilizations preserve and drop by . Type II stabilizations preserve and drop by . So, by applying sufficiently many stabilizations of appropriate types we make the Thurston–Bennequin numbers of all the components of smaller than the given . ∎

For any finite subset , we will use notation in the same sense as for rectangular diagrams:

If is not a rectangular diagram of a link, then is not a link, but is always a graph, i.e. a -dimensional CW-complex.

The following statement will be given in greater generality than necessary to establish Theorem 1. It will be used in full generality later in the proof of Theorem 2.

Proposition 5.

Let be a surface with corners, and two disjoint finite subsets of