A colored operad for string link infection

# A colored operad for string link infection

## Abstract

Budney constructed an operad that encodes splicing of knots and further showed that the space of (long) knots is generated over this splicing operad by the space of torus knots and hyperbolic knots. This generalized the satellite decomposition of knots from isotopy classes to the level of the space of knots. Infection by string links is a generalization of splicing from knots to links. We construct a colored operad that encodes string link infection. We prove that a certain subspace of the space of 2-component string links is generated over a suboperad of our operad by its subspace of prime links. This generalizes a result from joint work with Blair from isotopy classes of string links to the space of string links. Furthermore, all the relations in the monoid of 2-string links (as determined in our joint work with Blair) are captured by our infection operad.

string links
infection
operads
spaces of links
spaces of knots
\givenname

John \surnameBurke \givennameRobin \surnameKoytcheff \subjectprimarymsc200057M25 \subjectprimarymsc200018D50 \subjectprimarymsc200055P48 \subjectprimarymsc200057R40 \subjectprimarymsc200057R52 \arxivreference1311.4217 \arxivpasswordxufup

## 1 Introduction

This paper concerns operations on knots and links, particularly infection by string links. Classically, knots and links are considered as isotopy classes of embeddings of a 1-manifold into a 3-manifold, such as , , or . Instead of considering just isotopy classes, we consider the whole space of links, that is the space of embeddings of a certain 1-manifold into a certain 3-manifold. We also consider spaces parametrizing the operations and organize all of these spaces via the concept of an operad (or colored operad). The operad framework is in turn convenient for studying spaces of links and generalizing statements about isotopy classes to the space level. Finding such statements to generalize was the motivation for recent work of the authors and R Blair on isotopy classes of string links [1].

Our work closely follows the work of Budney. Budney first showed that the little 2-cubes operad acts on the space of (long) knots, which implies the well known commutativity of connect-sum of knots on isotopy classes. He showed that is freely generated over by the space of prime knots, generalizing the prime decomposition of knots of Schubert from isotopy classes to the level of the space of knots [2]. Later, he constructed a splicing operad which encodes splicing of knots. He showed that is freely generated over a certain suboperad of by the subspace of torus and hyperbolic knots, thus generalizing the satellite decomposition of knots from isotopy classes to the space level [4].

Infection by string links is a generalization of splicing from knots to links. This operation is most commonly used in studying knot concordance. One instance where string link infection arises is in the clasper surgery of Habiro [15], which is related to finite-type invariants of knots and links. In another vein, Cochran, Harvey, and Leidy observed that iterating the infection operation gives rise to a fractal-like structure [9]. This motivated our work, and we provide another perspective on the structure arising from string link infection. We do this by constructing a colored operad which encodes this infection operation. We then prove a statement that decomposes part of the space of 2-component string links via our colored operad.

Splicing and infection are both generalizations of the connect-sum operation. The latter is always a well defined operation on isotopy classes of knots, but if one considers long knots, it is even well defined on the knots themselves. This connect-sum operation (i.e., “stacking”) is also well defined for long (a.k.a. string) links with any number of components. Thus we restrict our attention to string links.

### 1.1 Basic definitions and remarks

Let and let be the unit disk with boundary.

###### Definition 1.1.

A -component string link (or -string link) is a proper embedding of disjoint intervals

 ∐cI↪I×D2

whose values and derivatives of all orders at the boundary points agree with those of a fixed embedding . For concreteness, we take to be the map which on the copy of is given by where . We will call the trivial string link. Another example of a string link is shown in Figure 1.

In our work [1], our definition of string links allowed more general embeddings, and the ones defined above were called “pure string links.” We choose the definition above in this paper because infection by string links behaves more nicely with this more restrictive notion of string link. (Specifically, it preserves the number of components in the infected link.)

The condition on derivatives is not always required in the literature.1 We impose it because this allows us to identify a -string link with an embedding which agrees with a fixed embedding outside of . Let denote the space of -string links, equipped with the Whitney topology. An isotopy of string links is a path in this space, so the path components of are precisely the isotopy classes of -string links. Often we will write for the space of long knots.

The braids which qualify as string links under Definition 1.1 are precisely the pure braids. There is a map from to the space of closed links in by taking the closure of a string link. When , this map is an isomorphism on . In other words, isotopy classes of long knots correspond to isotopy classes of closed knots. In general, this map is easily seen to be surjective on , but it is not injective on . For example, any string link and its conjugation by a pure braid yield isotopic closed links, and for , there are conjugations of string links by braids which are not isotopic to the original string link. We will sometimes write just “link” rather than “string link” or “closed link” when the type of link is either clear from the context or unimportant.

### 1.2 Main results

Our first main result is the construction of a colored operad encoding string link infection. An operad consists of spaces of -ary operations for all . Roughly, an operad acts on a space if each can parametrize ways of multiplying elements in . (We provide thorough definitions in Section 3.) A colored operad arises when different types of inputs must be treated differently. In our case, we have to treat string links with different numbers of components differently, so the colors in our colored operad are the natural numbers. This theorem is proven as Theorem 5.6 and Proposition 6.3.

###### Theorem 1.

There is a colored operad which encodes the infection operation and acts on spaces of string links for .

• When restricting to the color 1, the (ordinary) operad which we recover is Budney’s splicing operad, and the action of on is the same as Budney’s splicing operad action.

• For any , the operad obtained by restricting to is an operad which admits a map from the little intervals operad . The resulting -action on encodes the operation of stacking string links.

• On the level of , our infection operad encodes all the relations in the whole 2-string link monoid.

We then use our colored operad to decompose part of the space of string links. We rely on an analogue of prime decomposition for 2-string links proven in our joint work with R Blair [1], so we must restrict to . We consider a “stacking operad” , which is a suboperad of and which is homeomorphic to the little intervals operad. This operad simply encodes the operation of stacking 2-string links in , with the little intervals acting in the factor. The theorem below is proven as Theorem 6.8.

###### Theorem 2.

Let denote the submonoid of generated by those prime 2-string links which are not central. (By [1], this monoid is free.) Let be the subspace of consisting of the path components of that are in . Then is freely generated as a monoid over the stacking suboperad ; The generating space is the subspace consisting of those components in which correspond to prime string links.

### 1.3 Organization of the paper

In Section 2, we review the definition of string link infection.

In Section 3, we review the definitions of an operad and the particular example of the little cubes operad. We then give the more general definition of a colored operad.

In Section 4, we review Budney’s operad actions on the space of knots. This includes his action of the little 2-cubes operad, as well as the action of his splicing operad.

In Section 5, we define our colored operad for infection and prove Theorem 1. We make some remarks about our operad related to pure braids and rational tangles, and we briefly discuss a generalization to embedding spaces of more general manifolds.

In Section 6, we focus on the space of 2-string links. We prove Theorem 2, which decomposes part of the space of 2-string links in terms of a suboperad of our infection colored operad. We conclude with several other statements about the homotopy type of certain components of the space of 2-string links.

Notation:

• means

• means the restriction of to

• denotes the closure of ; denotes the interior of

• denotes the equivalence class represented by an element ; denotes the equivalence class of a tuple .

### 1.4 Acknowledgments

The authors thank Tom Goodwillie for useful comments and conversations. They thank Ryan Budney for useful explanations and especially for his work which inspired this project. They thank Ryan Blair for useful conversations and for invaluable contributions in their joint work with him, on which Theorem 2 depends. They thank a referee for a careful reading of the paper and useful comments. They thank Connie Leidy for suggesting the rough idea of this project. They thank Slava Krushkal for suggesting terminology and for pointing out the work of Habiro. Finally, they thank David White for introducing the authors to each other. The second author was supported partly by NSF grant DMS-1004610 and partly by a PIMS Postdoctoral Fellowship.

## 2 Infection

Infection is an operation which takes a link with additional decoration together with a string link and produces a link. This operation is a generalization of splicing which in turn is a generalization of the connect-sum operation. Infection has been called multi-infection by Cochran, Friedl, and Teichner [8], infection by a string link by Cochran [7] and tangle sum by Cochran and Orr [10]. Special cases of this construction have been used extensively since the late 1970’s, for example in the work of Gilmer [14]; Livingston [22]; Cochran, Orr, and Teichner [11, 12]; Harvey [16]; and Cimasoni [6]. The operad we define in this paper will encode a slightly more general operation than the infection operation that has been defined in previous literature. This section is meant to inform the reader of the definition in previous literature and provide motivation for the infection operad.

### 2.1 Splicing

Consider a link and a closed curve such that bounds an embedded disk in ( is unknoted in ) which intersects the link components transversely. Given a knot , one can create a new link , with the same number of components as , called the result of splicing by at . Informally, the splicing process is defined by taking the disk in bounded by ; cutting along the disk; grabbing the cut strands; tying them into the knot (with no twisting among the strands) and regluing. The result of splicing given a particular , and is show in Figure 2. Note that if simply linked one strand of then the result of the splicing would be isotopic to the connect-sum of and .

Formally, is arrived at by first removing a tubular neighborhood, , of from . Note is a solid torus with embedded in its interior. Let denote the complement in of a tubular neighborhood of . Since the boundary of is also a torus, one can identify these two manifolds along their boundary. In order to specify the identification, we use the terminology of meridians and longitudes. Recall that the meridian of a knot is the simple closed curve, up to ambient isotopy, on the boundary of the complement of the knot which bounds a disk in the closure of the tubular neighborhood of the knot and has +1 linking number with the knot. Also recall that the longitude of a knot is the simple closed curve, up to ambient isotopy, on the boundary of the complement of the knot which has +1 intersection number with the meridian of the knot and has zero linking number with the knot.

The gluing of to is done so that the meridian of the boundary of is identified with the meridian of in the boundary of . Note that this process describes a Dehn surgery with surgery coefficient along where the solid torus glued in is . Thus, the resulting manifold will be a 3-sphere with a subset of disjoint embedded circles whose union is (the image of under this identification). Although the embedding of in depends on the identification of the surgered 3-manifold with , its isotopy class is independent of this choice of identification.

### 2.2 String link infection

Although there is a well studied generalization of the connect-sum operation from closed knots to closed links, there is no generalization of splicing by a closed link. There is, however, a generalization of splicing called infection by a string link, which we will now define. See the work of Cochran, Friedl, and Teichner [8, Section 2.2] for a thorough reference.

By an -multi-disk we mean the oriented disk together with ordered embedded open disks (see Figure 3). Given a link we say that an embedding of an -multi-disk into is proper if the image of the multi-disk, denoted by , intersects the link components transversely and only in the images of the disks as in Figure 3. We will refer to the image of the boundary curves of by .

Suppose is link, is the image of a properly embedded -multi-disk, and is an component string link. Then informally, the infection of by at , denoted by , is the link obtained by tying the collections of strands of that intersect the disks into the pattern of the string link , where the strands linked by are identified with the component of , such that the collection of strands are parallel copies of the component of . Figure 5 shows an example of this operation.

We now define this operation formally. Given a string link , let denote the complement of a tubular neighborhood of (the image of) in . In Figure 4 an example of a string link is shown with its complement to the right. The meridian of a component of a string link is the simple closed curve, up to ambient isotopy, on the boundary of the closure of the tubular neighborhood of the component which bounds a disk and has +1 linking number with the component. We call the set of such meridians the meridians of the string link. The longitude of a component of a string link is a properly embedded line segment , up to ambient isotopy, on the boundary of the closure of the tubular neighborhood of the component; it is required to have +1 intersection number with the meridian of that component, to have zero linking number with that component, and to satisfy and . We call the set of such longitudes the longitudes of the string link. In Figure 4 the meridians, , and longitudes, , are shown on the boundary of the complement. Note that the boundary of the complement of any -component string link is homeomorphic to a genus- orientable surface.

Let be a link, and let be a string link. Fix a proper embedding of a thickened -multidisk in . Formally the infection of by at is obtained by removing from and gluing in the complement of . Note that is the complement of a -component trivial string link (see Figure 5), and thus the boundary of is a genus- orientable surface. One identifies this boundary and the boundary of the complement of , , first by identifying with subset of the boundary of where is taken to be a subset of the boundary of where lives, is identified with and the components of the closure of and are identified so that the meridians and longitudes of are identified with the meridians and longitudes of .

We claim that the resulting manifold is containing a link (the image of under this identification). The resulting manifold is homeomorphic to because

 S3∖Int((D∖⊔iφ(D1))×I)∪(D2×I)∖N(L) = (S3∖D×I)∪((⊔i(φ(Di)×I))∪(D2×I)∖N(L)) ≅ S3

where the last homeomorphism follows form the observation that the previous space is the union of two 3-balls. Again, the specific embedding of will depend on the choice of homeomorphism, but all choices will yield isotopic embeddings.

## 3 Operads

We start by reviewing the definitions of an operad , and an action of on (a.k.a. an algebra over ). We then proceed to colored operads. Technically, the definition of a colored operad subsumes the definition of an ordinary operad, but for ease of readability, we first present ordinary operads. Readers familiar with these concepts may safely skip this Section.

### 3.1 Operads

Operads can be defined in any symmetric monoidal category, but we will only consider the category of topological spaces. In this case, the rough idea is as follows. An algebra over an operad is a space with a multiplication , and the space parametrizes ways of multiplying elements of , i.e., maps . In other words, captures homotopies between different ways of multiplying the elements, as well as homotopies between these homotopies, etc. Thus an element of is an operation with inputs and one output. This can be visualized as a tree with leaves and a root, and in fact, free operads are certain spaces of decorated trees. For a more detailed introduction, the reader may wish to consult the book of Markl, Shnider, and Stasheff [23], May’s book [24], or the expository paper of McClure and Smith [25].

###### Definition 3.1.

An operad (in the category of spaces) consists of

• a space for each with an action of the symmetric group

• structure maps

 O(n)×O(k1)×...×O(kn)→O(k1+...+kn) (1)

such that the following three conditions are satisfied:

• Associativity: the following diagram commutes:

• Symmetry: Let denote the diagonal action on the product coming from the actions of on and on by permuting the factors. For a partition of a natural number , let denote the “block permutation” induced by and the partition .

We require that the following composition agrees with the map (1):

We also require that for for , the following diagram commutes:

• Identity: There exists an element (i.e., a map ) which induces the identity on via

 O(1)×O(k) →O(k) (1,L) ↦L

and which induces the identity on via

 O(n)×O(1)×O(1)×...×O(1) →O(n) (L,1,1,...,1) ↦L.

Some authors define the structure maps via operations, i.e., plugging in just one operation into the input, as opposed to operations into all inputs. These maps can be recovered from the above definition by setting for all and using the identity element in .

###### Definition 3.2.

Given an operad , an action of on (also called an algebra over ) is a space together with maps

 O(n)×Xn→X

such that the following conditions are satisfied:

• Associativity: The following diagram commutes:

• Symmetry: For each , the action map is -invariant, where acts on by definition, on by permuting the factors, and on the product diagonally. In other words, the action map descends to a map

 O(n)×ΣnXn→X
• Identity: The identity element together with the map

 O(1)×X→X

induce the identity map on , i.e., the map takes .

### 3.2 The little cubes operad

Our infection colored operad extends Budney’s splicing operad, which in turn was an extension of Budney’s action of the little 2-cubes operad on the space of long knots. Thus the little 2-cubes operad is of interest here.

###### Definition 3.3.

The little -cubes operad is the operad with the space of maps

 (L1,...,Ln):∐nIj↪Ij

which are embeddings when restricted to the interiors of the and which are increasing affine-linear maps in each coordinate. The structure maps are given by composition:

 Cj(n)×Cj(k1)×...×Cj(kn) →Cj(k1+...+kn) (L1,...,Ln),(L11,...,L1k1),...,(Ln1,...,Lnkn) ↦(L1∘(L11,...,L1k1),...,Ln∘(Ln1,...,Lnkn))

Note that for all , the multiplication induced by choosing (any) element in is commutative up to homotopy, which can be seen via the same picture that shows that is abelian for .

### 3.3 Colored Operads

Now we present the precise definitions of a colored operad and an action of a colored operad on a space. This generalization of an operad is necessary to generalize Budney’s operad from splicing of knots to infection by links.

###### Definition 3.4.

A colored operad (in the category of spaces) consists of

• a set of colors

• a space for each -tuple together with compatible maps for each

• (continuous) maps

 O(c1,...,cn;c)×O(d1,1,...,d1,k1;c1)×...×O(dn,1,...,dn,kn;cn)→O(d1,1,...,dn,kn;c)

where the maps satisfy the following three conditions:

• Associativity: The map below is the same regardless of whether one first applies the structure maps to the first two factors or the last two factors on the left-hand side:

• Symmetry: The following diagram below commutes. The vertical map is induced by in both the first factor and the last factors, and is the block permutation induced by and the partition .

We also require that, for , , the following diagram commutes:

• Identity: For every , there is an element which together with

 O(c;c)×O(c1,...,cn;c)→O(c1,...,cn;c)

induces the identity map on . We also require that the elements together with

 O(c1,...,cn;c)×O(c1;c1)×...×O(cn;cn)→O(c1,...,cn;c)

induce the identity map on .

The colors can be thought of as the colors of the inputs, while is the color of the output. A colored operad with is just an operad, where is . Sometimes, for brevity, we write “operad” to mean “colored operad.”

Note that if we have a colored operad with colors and a subset , we can restrict to another colored operad consisting of just the spaces with (and the same structure maps as ).

###### Definition 3.5.

Given a colored operad , an action of on (also called an -algebra ) consists of a collection of spaces together with maps

 O(c1,...,cn;c)×Ac1×...×Acn→Ac (2)

satisfying the following conditions:

• Associativity: The following diagram commutes:

• Symmetry: For each , the following diagram commutes, where the vertical map is induced by the -action and permuting the factors of :

• Identity: The map induced by together with is the identity on .

If we have a subset , the restriction colored operad acts on the collection of spaces .

###### Example 3.6.

A planar algebra as in the work of Jones [20] is an algebra over a certain colored operad. In fact, planar diagrams form a colored operad called the planar operad . The colors are the even natural numbers, and is the space of diagrams with holes, strands incident to the -th boundary circle, and strands incident to the outer boundary circle. If denotes the space of tangle diagrams in with endpoints on , then the collection is an example of an algebra over (a.k.a. a planar algebra).

## 4 A review of Budney’s operad actions

### 4.1 Budney’s 2-cubes action

The operation of connect-sum of knots is always well defined on isotopy classes of knots. If one considers long knots, one can further define connect-sum (or stacking) of knots themselves, rather than just the isotopy classes. That is, there is a well defined map

 #:K×K→K

where is the space of long knots. If one descends to isotopy classes, this operation is commutative, i.e., is homotopy-commutative. See Budney’s paper [2, p. 4, Figure 2] for a beautiful picture of the homotopies involved. This picture suggests that one can parametrize the operation by . Thus it suggests that the little -cubes operad acts on .

Budney succeeded in constructing such a 2-cubes action, but to do so, he had to consider a space of fat long knots

 EC(1,D2):={f:R1×D2↪R1×D2|supp(f)⊂I×D2}

where is defined as the closure of . The notation stands for (self-)embeddings of with cubical support. This space is equivalent to the space of framed long knots, but one can restrict to the subspace where the linking number of the curves and is zero; this subspace is then equivalent to the space of long knots.

The advantage of is that one can compose elements. In the 2-cubes action on this space, the first coordinate of a cube acts on the factor in , while the second factor dictates the order of composition of embeddings. Precisely, the action is defined as follows. For one little cube , let be the embedding given by projecting to the last factor. Let be the embedding given by projecting to the first factor(s). Let be a permutation (thought of as an ordering of ) such that . The action

 C2(n)×EC(1,D2)n→EC(1,D2)

is given by

 (L1,...,Ln)⋅(f1,...,fn)↦Lxσ(n)∘fσ(n)∘(Lxσ(n))−1∘...∘Lxσ(1)∘fσ(1)∘(Lxσ(1))−1.

### 4.2 The splicing operad

In the above 2-cubes action, the second coordinate is only used to order the embeddings. Thus instead of the 2-cubes operad, one could consider an operad of “overlapping intervals” . An element in is intervals in the unit interval, not necessarily disjoint, but with an order dictating which interval is above the other when two intervals do overlap. Precisely, an element of is an equivalence class where each is an embedding and where . Elements and are equivalent if for all and if whenever and intersect, . It is not hard to see what the structure maps for the operad are (and they are given in Budney’s paper [4]). Budney then easily recasts his 2-cubes action as an action of the overlapping intervals operad .

The splicing operad (which we abbreviate for now as ) is formally similar to the overlapping intervals operad, in that consists of equivalence classes of elements with the same equivalence relation as for . In the splicing operad, however, is in , are embeddings , and all the are required to satisfy a “continuity constraint,” as follows. One considers as an element of which fixes 0. If one can think of as inner (or first in order of composition) with respect to . One wants the “round boundary” of not to touch , but for the operad to have an identity element, one needs to allow for to be flush around . The precise requirement needed is that for

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯imLi∖imLk∩Lk(∘I×∂D2)=∅.

Note that is a much “bigger” operad than . One can think of as the “starting (thickened long) knot” for the splicing operation and of the other as “hockey pucks” with which one grabs and ties up into knots. This gives a map

 SC(n)×Kn→K

which will define the action of the splicing operad on . To fully construct as an operad, one needs the operad structure maps, which also come from the map above. Roughly speaking, the structure maps are as follows. Given one splicing diagram with pucks and other splicing diagrams each with pucks (), put the splicing diagram into the puck by composing the “starting knots” and “taking the pucks along for the ride.” For a precise definition and pictures, the reader may either consult [4] or read the next Section, which closely follows Budney’s construction and subsumes the splicing operad.

## 5 The infection colored operad

###### Definition 5.1.

Fix for each a trivial -component fat string link

 ic:∐cI×D2↪I×D2.

with image denoted .

We will be more concerned with this image of the fixed trivial fat string link rather than the embedding itself.

A convenient way of choosing an is to fix an embedding and then take the product with the identity map on . For , we choose an embedding which takes the centers of the ’s to the points from our definition (1.1) of string links. Beyond that, we remain agnostic about this fixed embedding. For , we choose to be the identity map. This will recover Budney’s splicing operad from our colored operad when all the colors are 1.

Now we define the space of -component fat string links to be

 FSLc:={f:∐cI×D2↪I×D2|f agrees with ic on ∂I×D2}.

These are the spaces on which the infection colored operad will act. An element of is displayed in Figure 6. By our condition on the fixed trivial fat string link, we can restrict to the cores of the solid cylinders to obtain an ordinary string link as in Definition 1.1.

### 5.1 The definition of the infection colored operad

We now define our colored operad . We put , so each color is a positive natural number.

###### Definition 5.2 (The spaces in the colored operad I).

An infection diagram is a tuple with , , and an embedding (for ) satisfying a certain continuity constraint. The constraint is that if , then

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Li(I×D2)∖Lk(Sck)∩Lk(∘I×(D2∖∘Sck))=∅ (†)

where is the image of a fixed trivial string link, as in Definition 5.1. As in the splicing operad, we think of as a permutation in which fixes 0.

The space is the space of equivalence classes of infection diagrams, where and are equivalent if for all , and if whenever the images of and intersect, if and only if . ∎

Informally, the are like the hockey pucks in Budney’s splicing operad, and the permutation is a map that sends the order of composition to the index of . The difference is that instead of re-embedding a hockey puck into itself, we will re-embed the image of , a subspace of thinner inner cylinders, into the puck. Thus we keep track of the image of , and our pucks can be thought of as having cylindrical holes drilled in them, the holes with which we will grab the string link (or long knot) . Following a suggestion of V. Krushkal, we call the restrictions of the to “mufflers” (motivated by the picture for ).

The generalization of Budney’s continuity constraint to the constraint is the key technical ingredient in defining our colored operad. The need for this constraint is explained precisely in Remark 5.4 below. The rough meaning of this condition is that a muffler which acts earlier should be inside a hole of a muffler that acts later; in other words, the “solid part” of a higher (which remains after drilling out the trivial string link) should not intersect any part of a lower , where “higher” and “lower” are in the semi-linear ordering determined by . However, we must allow for the possibility of the boundaries of the mufflers intersecting in certain ways. Figure 8 displays the cross-section of a set of mufflers which satisfy constraint .

So far we haven’t finished defining the operad, since we haven’t defined the structure maps. We start by defining the action on the space of fat string links. Only after that will we define the structure maps and check that they form a colored operad and that the definition below is a colored operad action.

###### Definition 5.3 (The action of I on fat string links).

Consider and fat string links where . Let be the map obtained from by restricting the domain to and restricting the codomain to its image. We use the shorthand notation

 Lk⋅fk to denote the map Lk∘fk∘(Link)−1:Lk(Sck)→I×D2.

Then we define

 I(c1,...,cn;c)×FSLc1×...×FSLcn →FSLc by ([L0,L1,...,Ln,σ],f1,...,fn) ↦(Lσ(n)⋅fσ(n))∘...∘(Lσ(1)⋅fσ(1))∘L0.\qed
###### Remark 5.4.

Strictly speaking, each map is only defined on , so one might worry whether the above composition is well defined. We claim that the conditions on the support of the and the continuity constraint () guarantee that we can continuously extend each by the identity on .

In fact, first write

 ∂(imLinσ(k))=(∂I×∐ckD2)∪(I×∂∐ckD2).

Since each is the identity on the part of its domain (the “flat boundary”), the map is the identity on the part of .

The constraint says that

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯imL0∖imLinσ(k)∩Lσ(k)(∘I×∂(∐ckD2))=∅,

hence

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯imL0∖imLinσ(k)∩imLσ(k)⊆∂I×∐ckD2.

So the continuity constraint guarantees that we don’t need to worry about extending past the part of the boundary (the “round boundary”).

Hence this defines the composition on the whole image of . ∎

###### Definition 5.5 (The structure maps in I).

The structure maps

 I(c1,...,cn;c)×I(d1,1,...,d1,k1;c1)×...×I(dn,1,...,dn,kn;cn)→I(d1,1,...,dn,kn;c) (3)
 (J0,...,Jn,ρ)×(L1,0,...,L1,k1,σ1)×...×(Ln,0,...,Ln,kn,σn)↦((J⋅→L)0,(J⋅→L)1,1,...,(J⋅→L)n,kn,τ)

are defined as follows. (Here , which can be thought of as infection diagrams, and is just shorthand for the result on the right-hand side.) The “starting” fat string link is

 (J⋅→L)0 :=(◯ni=1Jρ(i)⋅Lρ(i),0)∘J0 :=(Jρ(n)∘Lρ(n),0∘(Jinρ(n))−1)∘...∘(Jρ(1)∘Lρ(1),0∘(Jinρ(1))−1)∘J0.

Given and , the puck is

 (J⋅→L)a,b:=(◯ni=ρ−1(a)+1Jρ(i)⋅Lρ(i),0)∘(Ja∘La,b)

Finally, the permutation associated to is given by

 τ−1(a,b):=τ−1(b+a−1∑i=1ki):=σ−1a(b)+ρ(a)−1∑i=1kρ(i).

In other words

 τ−1:(1,1),(1,2),...,(n,kn)↦(ρ−1(1),σ−11(1)),(ρ−1(1),σ−11