\mathfrak{sl}_{n} homotopy type for matched diagrams

# An sln stable homotopy type for matched diagrams

Dan Jones Department of Mathematical Sciences
Durham University
Andrew Lobb Department of Mathematical Sciences
Durham University
and  Dirk Schütz Department of Mathematical Sciences
Durham University
###### Abstract.

There exists a simplified Bar-Natan Khovanov complex for open 2-braids. The Khovanov cohomology of a knot diagram made by gluing tangles of this type is therefore often amenable to calculation. We lift this idea to the level of the Lipshitz-Sarkar stable homotopy type and use it to make new computations.

Similarly, there exists a simplified Khovanov-Rozansky complex for open 2-braids with oppositely oriented strands and an even number of crossings. Diagrams made by gluing tangles of this type are called matched diagrams, and knots admitting matched diagrams are called bipartite knots. To a pair consisting of a matched diagram and a choice of integer , we associate a stable homotopy type. In the case this agrees with the Lipshitz-Sarkar stable homotopy type of the underlying knot. In the case the cohomology of the stable homotopy type agrees with the Khovanov-Rozansky cohomology of the underlying knot.

We make some consistency checks of this stable homotopy type and show that it exhibits interesting behaviour. For example we find a in the type for some diagram, and show that the type can be interesting for a diagram for which the Lipshitz-Sarkar type is a wedge of Moore spaces.

AL and DS were both supported by EPSRC grant EP/M000389/1, DJ was supported by an EPSRC graduate studentship.
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## 1. Introduction

In [LS14a], Lipshitz-Sarkar construct a stable homotopy type associated to a knot. They show that the cohomology of this object recovers Khovanov cohomology111We use Khovanov cohomology here rather than Khovanov homology for reasons discussed in Subsection 1.2. The construction of this stable homotopy type proceeds along lines laid down by Cohen-Jones-Segal [CJS95]. Allowing ourselves a good measure of imprecision, the idea can be summarized as follows.

To a knot diagram of a knot Khovanov associated a combinatorial bigraded cochain complex [Kho00]. The cohomology of this complex (Khovanov cohomology) is an invariant of , exhibiting as its graded Euler characteristic a knot polynomial - the Jones polynomial - and is projectively functorial for knot cobordisms. These properties (as well as host of spectral sequences starting from Khovanov cohomology and abutting to various Floer-theoretic invariants of the knot ), invite one to think of Khovanov cohomology as a type of Floer homology.

Following this mental yoga further, one might think of the standard generators of the Khovanov cochain complex as being akin to critical points of a Floer functional (or, more simply, of a Morse function). Then the differential describes -dimensional moduli spaces of flowlines between those critical points. If one can make a good guess as to what the higher-dimensional spaces of flowlines might be, then one can follow the recipe of Cohen-Jones-Segal [CJS95] and associate to a knot diagram a stable homotopy type whose cohomology recovers Khovanov cohomology. Finally one hopes that what one has constructed is invariant under the Reidemeister moves.

There is of course much difficulty to be overcome in making the previous two paragraphs yield up an honest stable homotopy type invariant. In particular, the input to the Cohen-Jones-Segal machine is more complicated than we have made out, in fact it takes the form of a framed embedded flow category, and constructing such a thing takes the majority of the paper [LS14a]. Nevertheless, one should think of framed flow categories as being closely related to Morse theory.

Our motivation for writing this paper was to attempt to extend this construction to the case of Khovanov-Rozansky cohomology (for ) in a way that might enable us and others to make computations. The Lipshitz-Sarkar stable homotopy type should appear as the case of this extension. Since there is a notion of stabilization of these cohomologies as , something similar should be true for the stable homotopy types.

Given the input of a special type of knot diagram (a matched diagram) we show how to create a stable homotopy type for each whose cohomology recovers the Khovanov-Rozansky cohomology of the underlying knot. In the case we recover the Lipshitz-Sarkar stable homotopy type. As well as obtaining the ‘correct’ thing when , we give some consistency checks suggesting that the space we construct is the right space.

The stable homotopy type we define is eminently computable and indeed we give some interesting computations at the end of this paper. In fact, the construction even makes the Lipshitz-Sarkar stable homotopy type more computable for certain knots (for example for pretzel knots) since it reduces the number of objects in the corresponding flow category. We do not show in this paper that the stable homotopy type for is independent of the choice of diagram, but this is something that we shall return to in future work.

### 1.1. Statement of results

In Figure 1 we define an elementary tangle diagram of index . Given a link diagram , there is of course always a decomposition of into such elementary pieces - for example one piece of index for each crossing of . When a link diagram comes together with such a decomposition we shall write it as where and the decomposition is into elementary tangles where the th tangle has index .

###### Definition 1.1.

We shall refer to the data of a link diagram together with such a choice of decomposition into elementary tangles as a glued diagram .

From now on we shall consider all diagrams to be oriented. Given a choice of integer and a glued diagram (we shall write when we intend to ignore the decomposition), we shall define a framed embedded flow category . The objects of the flow category are cohomologically graded, and the category splits as a disjoint union of flow categories along an quantum integer grading.

There is a space associated to by the Cohen-Jones-Segal construction - this takes the form of a stable homotopy type. The cohomology of is bigraded by the underlying cohomological degree and quantum grading of the objects of .

Following suitable degree choices we have

###### Theorem 1.2.

The space agrees with the space associated to the diagram by the Lipshitz-Sarkar construction.

In the case we restrict our attention to matched diagrams.

###### Definition 1.3.

A glued diagram is called matched if each coordinate of is even and each elementary tangle in the decomposition of has oppositely oriented strands. A link admitting such a diagram is called bipartite.

###### Remark 1.4.

Note that if is a knot diagram then the evenness condition implies the orientation condition.

Khovanov cohomology is a categorification of the Jones polynomial, which arises from the fundamental representation of via the Reshetikhin-Turaev construction. The categorification of the corresponding polynomial for is Khovanov-Rozansky cohomology [KR08]. Again, this is a bigraded link invariant and we shall write it as for an abelian group. That one can have general -coefficients (as opposed to just complex coefficients) was noted first in the case by Khovanov in his foam categorification. The case of was only recently shown by Queffelec-Rose [QR14] to have a foam interpretation valid with arbitrary coefficents.

With suitable grading shifts we have

###### Theorem 1.5.

If is a matched link diagram then the cohomology with complex coefficents of is isomorphic to as a bigraded complex vector space.

The reason for restricting this theorem to complex coefficients is not so serious. If one wanted to prove the theorem for arbitrary coefficients one would have to verify Krasner’s theorem [Kra09] on simplified Khovanov-Rozansky cochain complexes for matched diagrams over the integers (Krasner’s result is over the complex numbers). There is ample computational evidence that indeed Krasner’s theorem will hold in this case, and such an extension using Queffelec-Rose’s foamy definition of Khovanov-Rozansky cohomology over the integers should not be difficult, although we do not undertake it here.

Although the flow category depends heavily on the choice of decomposition of into elementary tangles we show that

###### Proposition 1.6.

If is a matched link diagram and , then is independent of the decomposition of into elementary tangles, and so can be written .

Furthermore, we derive relatives of Reidemeister moves I and II for matched diagrams.

###### Proposition 1.7.

In Figures 2 and 3 we give matched diagrams and that differ locally. For such diagrams we have that . The corresponding relations for the mirror images also hold.

Hence we have a multitude of evidence that our definition for this restricted class of diagrams is giving us the ‘correct’ stable homotopy type: the construction gives the Lipshitz-Sarkar space for (see also Remark 3.16); the cohomology of the space is the correct thing (for general using complex coefficients, and for verified over the integers for a host of examples); the space constructed is independent of the decomposition of the matched diagram into elementary tangles; and the space is invariant under extended Reidemeister moves I and II.

Unfortunately, there does not exist a genuine Reidemeister calculus to move between different matched diagrams of the same bipartite knot. It is probably not the case, for example, that just the moves of Figures 2 and 3 suffice! Furthermore it would be preferable to have a construction of an stable homotopy type for all links (not just bipartite links) which takes as input a link diagram and is invariant under the Reidemeister moves. We shall return to this question in a later paper.

We include at the end of this paper a section of computations, undertaken both by hand and by computer. Of note is the detection of a summand in the stable homotopy type for the pretzel link . A summand has yet to be detected in the Lipshitz-Sarkar stable homotopy type, although this may be due only to lack of computations.

Furthermore, it has long been known that the Khovanov cohomology of a knot may be thin while the HOMFLYPT cohomology is thick. At the level of spaces one should ask then if there is an example of a knot whose stable homotopy type is a wedge of Moore spaces while its stable homotopy type is more interesting for some . Such an example is provided by the pretzel knot whose stable homotopy type induces non-trivial second Steenrod squares.

Finally, work by Baues and by Baues-Hennes [Bau95, BH91] reveals ways to detect combinatorially more stable homotopy types than Moore spaces, , , , and (and wedge sums of these spaces). This is applied in the case of the Lipshitz-Sarkar stable homotopy type to the torus knots and after finding glued diagrams of each formed of a relatively low number of elementary tangles.

### 1.2. Conventions

In this paper we try to follow accepted conventions as far as we can. One way in which we shall differ from usual (although not universal) practice is to refer to the invariants due to Khovanov and to Khovanov-Rozansky as cohomology theories rather than homology theories. Since the differential in the complexes constructed does indeed increase the -grading this makes sense. It is also consistent with the spaces defined by Lipshitz-Sarkar and by us since the associated cohomology theories are the usual singular cohomologies of the spaces themselves.

In Figure 4 we fix what we mean by a positive or negative oriented crossing.

In both Khovanov and in Khovanov-Rozansky cohomology, the cohomology of the positive trefoil is supported in non-negative cohomological degrees. However, in Khovanov cohomology the invariant is supported in positive quantum degrees while in Khovanov-Rozansky cohomology (even in the case) the invariant is supported in negative quantum degrees. This is a consequence of the grading assigned to in the underlying Frobenius algebra , which is taken to be for Khovanov-Rozansky cohomology and for Khovanov cohomology. This is unfortunate but we stick to this convention in the current paper.

We shall sometimes be a little imprecise about absolute cohomological and quantum gradings. Especially in proofs, the extra decorations with shifts (depending on the writhe) tends to obscure the argument. When we wish to be more precise we shall use the postscript following a complex to denote a shift by in the cohomological direction and by in the quantum direction.

### 1.3. Plan of the paper

We begin in Section 2 with a review of framed flow categories and how the Cohen-Jones-Segal construction associates stable homotopy types to such things. In Section 3 we describe how we build a flow category given the data of a glued link diagram together with a choice of integer . Section 4 considers the properties of the associated stable homotopy type. In particular, the stable homotopy type returned for agrees with the Lipshitz-Sarkar stable homotopy type, while for the stable homotopy type for a matched diagram has cohomology agreeing with Khovanov-Rozansky cohomology. Finally in this section we provide some indictions discussed in the introduction that the stable homotopy type is the ‘correct’ space for . In Section 5 we discuss how to compute the second Steenrod square both in general flow categories and adjusted to our specific situations. The paper closes with Section 6 in which we provide explicit computations of several interesting stable homotopy types.

## 2. Framed flow categories and the Cohen-Jones-Segal construction

### 2.1. Framed flow categories

To define flow categories we need a sharpening of a manifold with corners that goes back to [Jän68]. Recall that a smooth manifold with corners is defined in the same way as an ordinary smooth manifold, except that the differentiable structure is now modelled on the open subsets of .

So if is a smooth manifold with corners and is represented by , let be the number of coordinates in this -tuple which are . Denote by

 ∂iX = {x∈X|c(x)=i}

the codimension--boundary. Note that belongs to at most different connected components of . We call a smooth manifold with faces if every is contained in the closure of exactly components of . A connected face is the closure of a component of , and a face is any union of pairwise disjoint connected faces (including the empty face). Note that every face is itself a manifold with faces. We define the boundary of , , as the closure of .

###### Definition 2.1.

Let be non-negative integers and a smooth manifold with faces. An -face structure for is an ordered -tuple of faces of such that

1. .

2. is a face of both and for .

A smooth manifold with faces together with an -face structure is called a smooth -manifold.

If , we define

 X(a) = ⋂i∈{j|aj=0}∂iX

and note that this is an -manifold, where . If we interpret the empty intersection as .

There is an obvious partial order on such that for .

###### Definition 2.2.

Given an -tuple of non-negative integers, let

 Ed = Rd0×[0,∞)×Rd1×[0,∞)×⋯×[0,∞)×Rdn.

Furthermore, if , we denote .

We can turn into an -manifold by setting

 ∂iEd = Rd0×⋯×Rdi−1×{0}×Rdi×⋯×Rdn.

We will refer to this boundary part as the -boundary. In the case of we also refer to the set

 ∂i−aEd[a:b] = Rda×⋯×Rdi−1×{0}×Rdi×⋯×Rdb−1

as the -boundary, although strictly speaking this should be the -boundary.

###### Definition 2.3.

A neat immersion of an -manifold is a smooth immersion for some such that

1. For all we have .

2. The intersection of and is perpendicular for all in .

A neat embedding is a neat immersion that is also an embedding.

Given a neat immersion we have a normal bundle for each immersion as the orthogonal complement of the tangent bundle of in .

###### Definition 2.4.

A flow category is a pair where is a category with finitely many objects and is a function, called the grading, satisfying the following:

1. for all , and for , is a smooth, compact -dimensional -manifold which we denote by .

2. For with , the composition map

 ∘:M(z,y)×M(x,z)→M(x,y)

is an embedding into . Furthermore,

 ∘−1(∂iM(x,y)) = {∂iM(z,y)×M(x,z)for im
3. For , induces a diffeomorphism

 ∂iM(x,y) ≅ ∐z,|z|=|y|+iM(z,y)×M(x,z).

We also write if we want to emphasize the flow category. The manifold is called the moduli space from to , and we also set .

Note that whenever , as the empty set is the only negative dimensional manifold.

###### Example 2.5.

Let be a Morse function on a closed manifold , and let be a Morse-Smale gradient for , meaning that all stable and unstable manifolds of intersect transversally. We then define the Morse flow category as follows.

The objects are exactly the critical points of , with grading given by the index. If is a critical point, define the stable and unstable manifolds with respect to the positive gradient flow, so that

 Ws(p) = {x∈M|limt→∞γx(t)=p}

where is the flowline of with , and

 Wu(p) = {x∈M|limt→−∞γx(t)=p}.

Given two different critical points and , let

 ~M(p,q) = Ws(p)∩Wu(q)/R,

where acts on this intersection using the flow. Then can be embedded into for every , and it follows from the transversality condition that this is a smooth manifold of dimension . To get the moduli space we compactify this space by adding all the broken flowlines between and using [AB95, Lm.2.6].

###### Definition 2.6.

Let be a flow category and a sequence of non-negative integers with for all . A neat immersion of the flow category relative is a collection of neat immersions for all objects such that for all objects and all points we have

 ıx,y(p∘q) = (ız,y(p),0,ıx,z(q)).

The neat immersion is called a neat embedding, if for all with the induced map

 ıi,j:∐(x,y),|x|=i,|y|=jM(x,y)→Ed[j:i]

is an embedding.

###### Definition 2.7.

Let be a neat immersion of a flow category relative . A coherent framing of is a framing for the normal bundle for all objects , such that the product framing of equals the pullback framing of for all objects .

A framed flow category is a triple , where is a flow category, a neat immersion and a coherent framing of .

Given a framed flow category , we can associate a chain complex as follows. The -th chain group is the free abelian group generated by the objects with grading , and if are objects with , the coefficient in the boundary between and is the sign of the -dimensional compact moduli space obtained from the framing in . The condition of a coherent framing ensures that we get indeed a chain complex.

Dually we can also associate a cochain complex .

There are various ways to think of a framing of an immersed manifold. For our constructions it will be useful to think of a framing of an immersion as an immersion

 φ:M(a,b)×[−ε,ε]d|b|+⋯d|a|−1→Ed[|b|:|a|]

such that for all .

### 2.2. The associated stable homotopy type

As was briefly alluded to in the introduction, there is a process that allows one to construct, from a given framed flow category , a CW complex . This CW complex is constructed in such a way that its cellular cochain complex is isomorphic (after some grading shift) to the cochain complex obtained from . An outline of the construction of was first given by Cohen-Jones-Segal (inspired by Franks [Fra79]) in attempt to achieve a spectrum (or space-level refinement) for Floer homology. As expressed in [CJS95], their attempt was not entirely successful but they do outline a detailed recipe for constructing a CW complex from any given framed flow category; a recipe that was implemented successfully in [LS14a] to produce such a spectrum for Khovanov cohomology, namely the Lipshitz-Sarkar stable homotopy type. The immediate output of the Cohen-Jones-Segal machine is a CW complex that we shall define in this section. It should be noted that the Lipshitz-Sarkar stable homotopy type is defined as a (de-)suspension of this output where the input is a particular Khovanov flow category, constructed in [LS14a].

###### Definition 2.8.

Let be a framed flow category relative . For an arbitrary object in of degree , recall that for each object in of degree , we have a framed neat embedding

 ıa,b:M(a,b)×[−ε,ε]dn+⋯+dm−1→[−R,R]dn×[0,R]×⋯×[0,R]×[−R,R]dm−1

where is chosen to be large enough that all moduli spaces can be embedded in this way. Moreover, choose as in Definition 2.6 so that every object satisfies . The CW complex consists of one -cell (the basepoint) and one -cell for every object of defined as

 [0,R]×[−R,R]dB×⋯×[−R,R]dm−1×{0}×[−ε,ε]dm×{0}×⋯×{0}×[−ε,ε]dA−1.

Each cell is considered a subset of a different copy of the ambient space . The neat embedding can be used to identify particular subsets

 (1) M(a,b)×C(b)≅Cb(a)⊂∂nC(a)

in the following way:

 Cb(a)= [0,R]×[−R,R]dB×⋯×[−R,R]dn−1×{0}× ıa,b(M(a,b)×[−ε,ε]dn+⋯+dm−1)× {0}×[−ε,ε]dm×⋯×{0}×[−ε,ε]dA−1 ⊂∂C(a).

It will be useful to introduce notation for this identification by letting

 (2) Γa,b:M(a,b)×C(b)→∂nC(a)

be the identification . Let

 (3) C=dB+⋯+dA−1−B.

Then the attaching map for each cell is defined via the Thom construction for each embedding into simultaneously. That is, for each subset , the attaching map projects to (which carries trivialisation information), and sends the rest of the boundary to the basepoint.

The fact that this construction is well-defined is shown in [LS14a, Lemma 3.25] which also describes how the attaching maps give a natural isomorphism of chain complexes.

The isomorphism type of is shown to be independent of the choice of real numbers and in [LS14a, Lemma 3.25] and, by considering a one-parameter family of framed neat embeddings between two perturbations and of , it can be shown that the CW complexes and are isomorphic (also [LS14a, Lemma 3.25]). A choice of different , , and gives rise to a stably homotopy equivalent CW complex (see [LS14a, Lemma 3.26]) that is a suspension of the original CW complex a number of times.

This construction of is also shown to agree with the construction of [CJS95] in [LS14a, Prop3.27], and is referred to as the realisation of .

## 3. A flow category associated to a glued diagram

The framed flow category that we associate to a matched diagram will be obtained in a somewhat similar way as that associated to a diagram in [LS14a]. The first difference is that we shall require a different flow category than the cube flow category downstairs. Again this category is essentially obtained from a Morse product construction. In the case of an elementary tangle with two crossings the factor category requires three consecutively graded objects, with one morphism point between the first two, two morphism points between the last two, and the -dimensional moduli space given by an interval. Morse-theoretically, this category can be obtained as in Figure 5. For more general diagram decompositions, the construction is a little more involved.

### 3.1. The sock flow category

For any integer and one can define the Lens space as the quotient space of by the action of given via

 t⋅(z0,…,zN)=(e2πi/nz0,…,e2πi/nzN)

where represents a generator of . It is well known that for all and otherwise. We want to define a Morse function and a Morse-Smale gradient for it which is perfect with respect to coefficients.

Let be the well-known Morse function given by

 f([z0:⋯:zN])=N∑j=1j|zj|2,

where . Now let be given by , where is the quotient map

 p(z0,…,zN)=[z0:⋯:zN].

Then is a Morse-Bott function with critical manifolds given by

 Sj={(0,…,0,zj,0,…,0)∈S2N+1}

each of which is a circle and whose index is given by , for .

For let be given by

 gn(z)=zn+¯zn,

a -invariant Morse function with maxima and minima, where acts on by

 t⋅z=z⋅e2πi/n.

Using , we can modify to a -invariant Morse function on which has exactly critical points, and which induces a -perfect Morse function on . In fact, it induces a -perfect Morse function on any Lens space , but we will only need a particular one.

For denote by the -equivariant negative gradient flow of . We can extend the flow to radially. Then given by

 ((z0,…,zN),t)↦(ψtz0,e−tψtz1,e−2tψtz2,…,e−NtψtzN)∥(ψtz0,e−tψtz1,e−2tψtz2,…,e−NtψtzN)∥

is a -equivariant negative gradient flow for .

In particular, this induces a negative gradient flow on to the induced Morse function . It is easy to see that the critical points are given by

 p2j =[0,…,0,eπi/n,0,…,0] p2j+1 =[0,…,0,1,0,…,0]

for with the non-zero entry in the -th coordinate, and the stable and unstable manifolds intersect transversely. We can therefore form the Morse flow category.

It is clear that consists of two points, one will be framed and the other will be framed .

###### Lemma 3.1.

For and the open moduli spaces are a disjoint union of components, each of which is diffeomorphic to an open disc. Furthermore, the action of on given by

 (4) t⋅[z0,z1,…,zN]=[e2πi/nz0,z1,…,zN]

induces an action on moduli spaces which permutes the components.

###### Proof.

Points are of the form

 x=[0,…,0,zj,…,zk,0,…,0]

with . Furthermore, we can assume that for and for .

Also, if , then

 argzj∈(0,2π/n)∪(2π/n,4π/n)∪⋯∪((2n−2)π/n,2π)

and if , then . Therefore there are at least components depending on the component that is in. To see that these are all the components, and each component is a disc, consider the point

 [0,…,0,e(1−ε2)π/n/√2,0,…,0,eε1π/n/√2,0,…,0].

This point can be considered as an origin, and all points are in a straight line to this point or to the analogue point in a different component, and we can identify each component as a star-like open subset of Euclidean space, which shows that each component is diffeomorphic to an open disc.

The statement about the action on moduli spaces is clear, with the action being trivial for . ∎

In particular, we get for , where each corresponds to the trajectory .

The -dimensional moduli spaces are easily identified as intervals with boundaries pairing as with for and with . Similarly, consists of intervals with boundaries pairing as with for and with .

We will also need to know the -dimensional moduli spaces . As the boundary is determined by the lower dimensional moduli spaces between these objects, and we know the number of components by Lemma 3.1, we see that the moduli space are squares with corners given by , , and for , where we identifiy and . As we shall see later, we do not need to know other moduli spaces.

Given , we can define a flow category whose objects are given by , and the moduli spaces are given by

 M(k,j)=M(pk+1,pj+1)/(Z/n).

In particular, if . Here is chosen so that to ensure that all moduli spaces are defined.

For and we define

 M(k,j)=M(p−j+1,p−k+1)/(Z/n)

to get the flow category which is dual to . Note that this is a quotient of the Morse flow category of the Morse function .

For let be the Morse function given by

 G(x1,…,xm)=sgn(r1)g(x1)+⋯+sgn(rm)g(xm)

and the corresponding Morse-flow category, where . The group acts on by letting each coordinate in act on the corresponding coordinate in by (4). This action induces an action of on the moduli spaces of and we define the flow category as follows.

###### Definition 3.2.

The objects of are given by with if or if , for all . The grading is given by and the moduli spaces are given by

 M((a1,…,am),(b1,…,bm))=MG((p|a1|+1,…,p|am|+1),(p|b1|+1,…,p|bm|+1))/(Z/n)m.

Note that since we use the negative Morse function for coordinates with , we do not have to interchange the role of and in those coordinates.

###### Proposition 3.3.

Let and objects in . Then is a disjoint union of discs.

###### Proof.

From the construction of the flow category it is enough to show that the same holds for the flow category of the Morse function defined on the product of lens spaces. To see that this holds we use induction on . For it follows from the description of the Morse flow category given above, where all moduli spaces are disjoint unions of cubes. The induction step follows from Lemma 3.4 below. ∎

###### Lemma 3.4.

Let and be Morse functions on closed, smooth manifolds and , and let , be Morse-Smale gradients for the respective Morse function. Let be the Morse flow category of the Morse function given by with Morse-Smale gradient . If , are critical points of with and , are critical points of with , then is PL-homeomorphic to

• if and .

• if and .

• if and .

Note that for and we usually do not get a diffeomorphism between and , as there will usually be more corners in .

###### Proof.

The cases where or are easy to see, so we will focus on the case where and . The proof is by induction on with the root case being trivial.

To simplify our notations we will drop the from the moduli spaces. We will write for the critical point of and similarly with other combinations of critical points of and .

Let be a critical point of with and a critical point of with . Since is a flow category, the boundary of is

If or , this picture simplifies, and the following arguments simplify also.

Note that , and the same holds for . We want to show that is a cylinder between these two boundary parts. To distinguish these parts more easily, we write

 M(a,b)⊠M(p,q)⊂M(ap,bq)

for to indicate that these are the broken flow lines that first go from to , and then from to . Hence we also write .

By induction hypothesis, we have

and

and we can think of these two boundary parts as combining to a cylinder between

 M(a,c)⊠M(c,b)⊠M(p,q)⊂M(a,b)⊠M(p,q)

and

 M(p,q)⊠M(a,c)⊠M(c,b)⊂M(p,q)⊠M(a,b)

with in the middle.

Similarly, is a cylinder between

 M(a,b)⊠M(p,r)⊠M(r,q)⊂M(a,b)⊠M(p,q)

and

 M(p,r)⊠M(r,q)⊠M(