On Operadic Actions on Spaces of Knots and 2-Links

In the present work, we realize the space of string 2-links $\mathcal{L}$ as a free algebra over a colored operad denoted $\mathcal{SCL}$ (for"Swiss-Cheese for links"). This result extends works of Burke and Koytcheff about the quotient of $\mathcal{L}$ by its center and is compatible with Budney's freeness theorem for long knots. From an algebraic point of view, our main result refines Blaire, Burke and Koytcheff's theorem on the monoid of isotopy classes of string links. Topologically, it expresses the homotopy type of the isotopy class of a string 2-link in terms of the homotopy types of the classes of its prime factors.


Introduction
Motivation and Context. The study of knots and links is a vast subject that emerged in the late nineteenth century and saw several renewals in the past thirty years. It is subject to many different approaches, being at the crossroads of topology, geometry, algebra, combinatorics and physics. The central theme in classical knot theory is the study of the isotopy classes of knots, i.e the isotopy classes of embeddings EmbpS 1 , S 3 q. They realize as the space of components π 0 EmbpS 1 , S 3 q. A Date: January 14, 2022.
common method is to try to split the knots into simpler pieces. Two ways of performing such a decomposition have proved themselves particularly fruitful: the prime decomposition and the satellite decomposition. The former splits a knot as the connected sum of other knots called its prime factors. The connected sum is a binary operation on π 0 EmbpS 1 , S 3 q denoted #. It endows the isotopy classes with a unital commutative monoidal structure. Intuitively, the product k 1 #k 2 is the knot obtained by cutting open k 1 and k 2 and closing them up into a single knot. This decomposition is fairly well understood thanks to a theorem of Schubert from [Sch49] stating that the monoid π 0 EmbpS 1 , S 3 q is freely generated by the isotopy classes of prime knots. There are infinitely many prime knots and some of them are very different in nature. This is why it is often useful to further decompose the prime knots as satellites of simpler knots. The satellite construction also originates in Schubert's work. It consists of a wide family of operations one can carry out on several knots at a time. Its rigourous definition is a bit complex but recently exposed in [Cro04]. Although they are quite complicated, the satellite operations have the advantage of generating the whole space of knots from a fairly small and classifyable class of knots. Namely, it is shown in [Bud06] that every knot can be obtained via successive satellite operations on hyperbolic and torus knots.
This paints the picture of a duality where one can choose to split a knot according to two different processes: a simpler one with potentially complicated primes, or a complex one with more elementary pieces. A similar story can be told for links, i.e the isotopy classes of embeddings EmbpS 1 >¨¨¨> S 1 , S 3 q, but the connected sum # does not formalize as well. This comes from the fact that once the components of a link are cut open, there is no canonical way to close them up. However, a step-by-step adaptation of the satellite construction works for links, and a decomposition theorem exists as well. Its elementary bricks are the hyperbolic and Seifert-fibered links.
Nowadays, it is more common to study not only the space of components π 0 EmbpS 1 >¨¨¨>S 1 , S 3 q, but the full homotopy type of the spaces of knots and links. To adapt the decomposition approach described above, one needs to define an analogue of the connected sum and satellite operations on the space level, i.e directly on the embeddings and not between isotopy classes. To rigourously carry out this task, new models for the spaces of knots and links have emerged, the long knots K and the string links L. Coupled with the language of operads, these new models enable one to extend the existing operations on π 0 to the space level. More precisely, the different types of operations one can carry out on knots and links can be encoded in the action of an operad on a space of embeddings. This new framework is due to Budney in [Bud07] and [Bud12] for the case of long knots. Namely, Budney constructs in [Bud07] an action of the little 2-cubes operad C 2 on a spaceK homotopy equivalent to K, in such a way that all the induced operations descend to the connected sum on π 0 . He builds in [Bud12] another action onK of a more intricate operad which he calls the splicing operad. These operations correspond to the satellite constructions in many ways. In the case of string links, Burke and Koytcheff build in [BK15] a complex operad called the infection operad. It is an adaptation of Budney's splicing operad and deals with the satellite operations between string links. The authors mentionned above not only prove the existence of such actions but also their freeness, refining the unique decomposition results on π 0 . It remains to find an operadic encoding of the connected sum of links, if possible leading to a free algebra.
Present Work. The present paper aims to fill in this gap in the case of 2-stranded string links. Unlike π 0 K, the monoid of isotopy classes of string 2-links is neither free nor commutative. Indeed, as explained in [BBK15], π 0 L contains invertible elements in the form of the pure braid group KB 2 . Together with three copies of π 0 K, these invertible elements generate the center of the monoid.
Burke and Koytcheff state a partial result in [BK15] by building a free action of the little 1-cubes operad C 1 on a subspace of string 2-links that ignores the homotopy center. They mention as an open problem the extension of such a structure to the whole space of string 2-links. Our main result provides an answer to this particular question. For this purpose, we introduce a four-colored operad SCL (standing for "Swiss-Cheese for links") with set of colors S " to, Ò, Ó, Ùu. An SCL-algebra is in particular a family of spaces pX, A Ò , A Ó , A Ù q where X is a C 1 -algebra and each A s is a C 2 -algebra acting on X, s P tÒ, Ó, Ùu. One can think of the A s as independent parts of the homotopy center of X. As in the case of Budney's C 2 -action on long knots, we consider a spaceL homotopy equivalent to L to prove a first result which can be summarized as follows: Theorem (Theorem 3.9). The family pL,K,K,Kq is a SCL-algebra. In particular, the family pL, K, K, Kq is homotopy equivalent to an explicit SCL-algebra.
The structure so obtained is compatible with Budney's action on long knots and Burke and Koytcheff's action on their subspace of non-central string links. It provides a complete understanding of the connected sum of string 2-links. As expected, we also prove a freeness result, refining the structure Theorem for the monoid of isotopy classes proved in [BBK15]. In order to do this, we discard the invertible elements by splitting π 0L as a product π 0L 0ˆK B 2 and prove: Theorem (Theorem 4.11). The quadruplet of spaces spaces pL 0 ,K,K,Kq is homotopy equivalent as a SCL-algebra to the free SCL-algebra generated by prime knots and links.
In addition to these algebraic considerations, this result has a homotopical significance as it expresses the homotopy type of the isotopy class of a string 2-link as a function of the homotopy types of the classes of its prime factors. This reduces the computation of the homotopy type of the whole L to figuring out the homotopy types of the components of the primes. As mentionned above, the latter can be further decomposed thanks to Burke and Koytcheff's infection operad defined in [BK15].
Organization of the paper. We define in a first section the different spaces of embeddings at stake here: long knots, string links and their fattened versionsK andL. The second section sets up the operadic framework we use. We recall in particular the notion of colored operad and discuss the resulting algebras. We introduce along the way the operad that will appear in our main result, the Swiss Cheese operad for links SCL. The third section aims to define various operadic actions on the spaces of knots and links. We recall the constructions of Budney, Burke and Koytcheff's actions and unify them in a single action of SCL on string 2-links. Finally, the fourth section proves the freeness result sketched above using low dimensional and homotopical tools.
Upcoming Projects. Our main statements concern the space of string links on 2 strands. One might naturally wonder what happens in the k-stranded case for some k ą 2. The conjecture that Theorems 3.9 and 4.11 have adaptations to arbitrary string links seems reasonable, since most of the techniques used Section 4.1 naturally generalize to the k-stranded case. However, lots of difficulties arise, even at the level of isotopy classes. Theorem 4.11 generalizes Blair, Burke and Koytcheff's explicit model for the monoid π 0 L, but it does not provide an alternative proof for it. Acutally, Blair, Burke and Koytcheff's result is used in the very first line of the proof of our Theorem 4.11. Thus, if one wants to adapt Theorem 4.11 to the k-stranded case, some preliminary work on the monoid of isotopy classes of string links on k strands is necessary. A key point is the understanding of its center. The commutation between string links on k strands has already been characterized in [BBK15], but there are other relations among prime links that remain to be understood. For example, the invertible k-stranded string links are the pure braids on k strands, which already admit a wide family of fairly complex relations amongst themselves. Moreover, these units are not central anymore, which makes it harder to study prime decompositions. A solution to these difficulties could be to accept a less explicit construction for the replacement of SCL, maybe a definition by induction on k. The operad for k-stranded string links would rely on a large set of colors, and would restrict to the operad for pk´1q-stranded string links on some subcollections of colors. The existence and freeness of an action could then be easier to prove, but the difficulty is only shifted towards understanding these potentially massive operads.
Another interesting question concerns the Goodwillie-Weiss' calculus, introduced to study embedding spaces in [Wei99,GW99]. In the context of knots and links, this theory gives rise to two towers of fibrations tT kK u and tT kL u, converging to the so-called polynomial approximations T ∞K and T ∞L , respectively. Unfortunately, the natural applications ιK :K Ñ T ∞K and ιL :L Ñ T ∞L are not weak equivalences, but they preserve a lot of homotopical information. In particular, we know from [BCKS14] that the mapK Ñ T kK is a finite type-pk´1q knot invariant and it has been conjectured that T kK is actually the universal finite type-pk´1q invariant. This conjecture is already proved rationally in Volic's thesis [Vol04]. Moreover, the polynomial approximations can be simplified and identified to homotopy totalisations using the multiplicative Kontsevich operad K 3 obtained as a compactification of configurations of points in R 3 . Briefly speaking, one has the identificationsK T ∞K hoT otpK 3˝S Op3qq, where K 2 3 pkq " K 3 p2kq is a shifted version of the Kontsevich operad. The applications µK and µL have been proved to be weak equivalences by Sinha in [Sin06] and Munson-Volic in [MV14], respectively. We know that the spacesK, T ∞K and hoT otpK 3˝S Op3qq are C 2 -algebras by [Bud07], [BdBW13] and [DKon], respectively. However, it is still unknown if ιK and µK are C 2 -algebra maps. All these questions can be extended to the colored case. From the present work, the family pL,K,K,Kq is equipped with an explicit SCL-algebra structure. We believe that similar structures exist for the families pT ∞L , T ∞K , T ∞K , T ∞K q, phoT otpK 2 3˝S Op3q, hoT otpK 3˝S Op3q, hoT otpK 3˝S Op3q, hoT otpK 3˝S Op3qq, and that the "zig-zag" of morphisms induced by ιK, µK, ιL and µL between the corresponding families are morphisms of SCL-algebras.
Acknowledgements. The authors are indebted to Thomas Willwacher and Robin Koytcheff for answering numerous questions and for their comments about the first version of the present paper. The two authors are also grateful to Dev Sinha and Victor Turchin, for discussions leading us to the problem solved in the paper. Finally, the authors acknowledge the ETH Zürich for generous support. The second author is partially supported by the grant ERC-2015-StG 678156 GRAPHCPX.

General Framework and Notations
We setup here the global framwork we work in as well as some notations that might not be completely standard.
Topological Spaces. By spaces, we understand compactly generated Hausdorff spaces. They form a full subcategory of topological spaces that we denote Top by slight abuse of notation. Many useful properties of Top have been introduced by Steenrod in [Ste67]. The standard Quillen model structure has then been adapted for it by Hovey in [Hov99]. It is a convenient category in the sense that the natural curryfication map ToppXˆY, Zq -ToppX, ToppY, Zqq is a homeomorphism for any three spaces X, Y and Z in Top. The need to restrict ourselves to such a subcategory arises from the following fact: when defining an action of an algebraic structure that it also a topological space A on a space X, one can ask for the continuity of either AˆX Ñ X or A Ñ ToppX, Xq. The homeomorphism above gives the equivalence between these two approaches and enables one to go back and forth between both frameworks. This will be useful when dealing with operadic actions.
Smooth Manifolds. When discussing manifolds, we think of usual (possibly bordered) C ∞ manifolds. We write I " r0, 1s for the unit interval and J " r´1, 1s for the 1-dimensional unit disc. The set of C ∞ maps between two manifolds M and N is denoted C ∞ pM, N q and topologized with the usual C ∞ -topology described in [Hir94]. The space of embeddings, immersions, submersions or diffeomorphisms between manifolds are topologized as subspaces of the latter. This turns diffeomorphism groups into topological groups and makes every composition map continuous.
Operations on Maps. Let f : A Ñ X and g : B Ñ Y be maps between spaces. We use the following notations: ‚ fˆg is the map between products AˆB Ñ XˆY , ‚ pf, gq is the map A Ñ XˆY when A " B, ‚ Aˆn is the product of n copies of A and fˆn is the map Aˆn Ñ Bˆn, ‚ B >n is the coproduct of n copies of B and f >n is the map A >n Ñ B >n .

Embedding Spaces
This section aims to review the construction of various spaces of embeddings, namely spaces of knots and 2-links. We start by recalling the definition of the usual space of knots and introduce three variations: the long knots K, the framed long knots ECp1, D 2 q and the fat long knotsK. These new versions are meant to ease algebraic and homotopical manipulations. We also discuss the classical monoid structure on the space of knots, its interactions with the newly defined spaces and finally adapt these constructions to 2-stranded links.
1.1. Knot Spaces. The first instance of a space of knots arises as the space of embeddings EmbpS 1 , S 3 q. Its components π 0 EmbpS 1 , S 3 q are the isotopy classes of knots in the 3-sphere and are the central object of study in knot theory. The class of the standardly embedded circle S 1 ãÑ R 3 Ă S 3 is called the trivial knot or unknot. Given two (isotopy classes of) knots k 1 and k 2 , one can define the connected sum k 1 #k 2 in various ways, as done for instance in [Cro04]. Intuitively, k 1 #k 2 is obtained by cutting open k 1 and k 2 and closing them back into a single knot. An example is provided in Figure 1. This operation turns out to be associtative, commutative and unital with the unknot as unit. This turns π 0 EmbpS 1 , S 3 q into a commutative monoid. The non-trivial elements k which admit no non-trivial factorization k " k 1 #k 2 are called prime. They are in a sense the most elementary knots. However, there are infinitely many of them and a further decomposition developped in [Bud06] suggests that they form a fairly wide class of knots. The monoid structure on π 0 EmbpS 1 , S 3 q is completely understood thanks to a theorem of Schubert: [Sch49]). The monoid π 0 EmbpS 1 , S 3 q is the free commutative monoid generated by prime knots.
# " Figure 1. Illustration of the connected sum of two knots.
We now introduce long knots. They are a mild variation of usual knots for which the connected sum is easier to deal with. Let ı : R Ñ R 3 , with ıptq " pt, 0, 0q be the standard embedding of the real line in R 3 . Definition 1.2. A long knot is an embedding R ãÑ R 3 that agrees with the standard embedding ı outside of J " r´1, 1s and maps the interior of J in the interior of JˆD 2 Ă R 3 . The space of long knots is denoted K. One can alternatively think of a long knot as a proper embedding J ãÑ JˆD 2 whose values and derivatives at BJ match those of ı.
With these conditions on the embeddings, it is natural to define a binary stacking operation between long knots as follows. Let L, R : R 3 Ñ R 3 be the maps sending px, y, zq to p x´1 2 , y, zq and p x`1 2 , y, zq, respectively. Given k 1 and k 2 in K, we define the concatenation of k 1 and k 2 as the long knot that restricts to t Þ Ñ L˝k 1 p2t´1q on r´1, 0s and to t Þ Ñ R˝k 2 p2t`1q on r0, 1s. This operation and its commutativity up to homotopy are illustrated in Figure 2. We still denote this operation # as it is the analogue of the connected sum in the following sense. Each long knot is linear outside of J and can therefore be extended to an embedding S 1 ãÑ S 3 by compactifying the domain and codomain. This specifies an inclusion K ãÑ EmbpS 1 , S 3 q which turns out to be a bijection on π 0 . It is an easy verification to check that the concatenation of two knots is sent to their connected sum. The isotopy classes π 0 K therefore inherit a monoid structure. When P denotes the collection of long knots which are prime, Schubert's Theorem applies and gives: Theorem 1.3. The monoid π 0 K is the free commutative monoid on the basis π 0 P. Figure 2. Illustration of the commutativity in the monoid π 0 K.
We now define framed long knots. The latter are meant to approximate the long knots defined above while being composable. The idea is to thicken the strand R into a long tube RˆD 2 . This definition is due to Budney and originates in [Bud07]. We define: Definition 1.4 (Budney, [Bud07]). A framed long knot is an embedding RˆD 2 Ñ RˆD 2 that restricts to the identity outside of JˆD 2 . The space of framed long knots is denoted ECp1, D 2 q. When supppf q denotes the support of an embedding f : RˆD 2 ãÑ RˆD 2 , i.e the closure of tpx, tq P RˆD 2 , f pt, xq ‰ pt, xqu, the condition for f to lie in ECp1, D 2 q can be reformulated as supppf q Ă JˆD 2 .
Note that this new space is still equipped with a stacking operation # defined just as in the case of long knots. It also still descends to an associative, commutative unital pairing on isotopy classes. There is a restriction map ECp1, D 2 q Ñ K; f Þ Ñ f |Rˆp0,0q that preserves the concatenation. But, it does not induce a bijection on π 0 . To see this, consider the diffeomorphism r of JˆD 2 that progressively performs a full turn rotation on the disc factor. One can parametrize r : pt, xq Þ Ñ pt, e iπpt`1q xq. This diffeomorphism can be isotoped about BJˆD 2 to be extended into an element of ECp1, D 2 q. Now, for any long knot k and f an extension in ECp1, D 2 q, each composite f˝r˝n also extends k but no two are isotopic. This produces infinitely many components in the fiber over k, which shows that the restriction map does not induce a bijection on π 0 . As of here, this twisting phenomenom is an unwanted byproduct of the thickenning process. We will get rid of it when definingK as an unframed subspace of ECp1, D 2 q. To do so, we need to quantify the framing of a knot, which we do via the framing number.
Definition 1.5. We define the framing number ωpf q of a framed long knot f P ECp1, D 2 q as the linking number lkpf |Rˆp0,0q , f |Rˆp0,1q q. Stricly speaking, the linking number is only defined between disjoint closed curves. We deal with this problem by identifying the curves above with their extension by compactification S 1 Ñ S 3 and isotoping f |Rˆp0,1q about the point at infinity to keep the disjointness.
Intuitively, the linking number counts the number of times a closed curve winds around an another. Here, we think of ωpf q as the number of times a curve on the surface of the knot wraps around the core f |Rˆp0,0q . This provides a way to quantify the framing of the elements of ECp1, D 2 q and we have ωpr˝nq " n. Another example is provided in Figure 3. Since the linking number can be computed by counting crossings in diagrams, it is easy to see that ω is additive with respect to the stacking product. It is also isotopy invariant and therefore descends to a morphism of monoids π 0 ECp1, D 2 q Ñ Z. We are finally able to defineK, our prefered model to approximate K. We simply set: Definition 1.6 (Budney, [Bud07]). The space of fat long knots is the subspaceK " ω´1p0q.
Fat long knots are stable under # thanks to the additivity of ω. The primes inK are denotedP. There still is a restriction mapK Ñ K which preserves this structure.K is a good approximation for K in the sense that: Proposition 1.7 (Budney, [Bud07]). The restriction mapK Ñ K is a homotopy equivalence.
In particular, we get an isomorphism on π 0 which enables us to tranfer Schubert's Theorem to fat long knots: Corollary 1.8. The monoid π 0K is the free commutative monoid on the basis π 0P .
1.2. Link Spaces. We now adapt these constructions for 2-links. Most of this generelization work has already been carried out by Burke and Koytcheff in [BK15]. The space of usual 2-links arises as EmbpS 1 > S 1 , S 3 q. There is no canonical version of a connected sum operation here, as there is no prefered strand in each link for one to merge. As in the case of long knots, the space of string links L deals with this problem by setting a framework where a stacking operation is naturally defined. Let ı 2 : R > R ãÑ R 3 be the embedding of two copies of the real line in R 3 mapping the first copy as t Þ Ñ pt, 0, 1 {2q and the other one as t Þ Ñ pt, 0,´1{2q. We refer to ı 2 as the standard embedding for links with two strands. We then define: Definition 1.9. A string 2-link is an embedding R > R ãÑ R 3 that agrees with ı 2 outside of J > J and maps the interior of J > J in the interior of JˆD 2 Ă R 3 . The space of string 2-links is denoted L. One can alternatively think of a string 2-link as a proper embedding J > J ãÑ JˆD 2 whose values and derivatives at BJ > BJ match those of ı 2 .
A binary stacking operation # can now be defined on L as in the case of long knots. It turns π 0 L into a monoid with unit ı 2 . A string 2-link is said to be prime if it is not invertible but cannot be factored without an invertible element. There is an injection L ãÑ EmbpS 1 > S 1 , S 3 q obtained by closing a truncated link f |J>J with two fixed smooth curves from p´1,˘1{2q to p1,˘1{2q as illustrated below. But, this inclusion does not induce a bijection on isotopy classes. Indeed, there are pairs of non-isotopic string 2-links that yield isotopic links once closed as shown in Figure 4. Therefore, studying string links slightly differs from usual link theory. Let us spend some time to investigate π 0 L. We first identify the invertible elements. There is a canonical map from the pure braid group on 2 strands to π 0 L sending a pure braid to its isotopy class as a string link. It is a morphism of monoids that only maps to units in π 0 L since braids form a group. This association is easily showed to be injective: the linking number map lk : L Ñ EmbpS 1 > S 1 , S 3 q Ñ Z descends to a left inverse when one identifies the pure braids on 2-strands with the integers in the natural way. This provides a whole collection of invertible elements and it turns out that every unit in π 0 L is of this form. Observe as well that these invertible links commute with every other link: an isotopy exhibiting this relation is suggested by Figure 7. Let now L 0 be the preimage of 0 through the linking number map lk. The injection of the braid group provides a section in the short exact sequence Thus π 0 L splits as π 0 L 0ˆZ and we can focus on the first factor. The monoid π 0 L 0 is not commutative but it contains several copies of π 0 K in its center. Indeed, consider the injective morphism ϕ Ò : π 0 K Ñ π 0 L 0 mapping a long knot k to the string link whose upper strand is knotted according to k and does not interact with the unknotted lower strand. An illustration of ϕ Ò is given in Figure 6. The image of ϕ Ò is a copy of π 0 K lying in π 0 L 0 , and one can build a similar morphism ϕ Ó by switching the roles of the strands. A third copy of the knot monoid can be found as follows. Consider the morphism ϕ Ù that sends a knot k to the link whose strands are parallel and unlinked but knotted according to k. This ϕ Ù maps to a third copy of π 0 K in π 0 L 0 . The images of any two of these morphisms intersect only in ı 2 so that π 0 Kˆ3 actually lives in π 0 L 0 . Every string link in the image of a ϕ s commutes with any other link, s P tÒ, Ó, Ùu. The structure theorem for string 2-links proved by Blair, Burke and Koytcheff in [BBK15] actually shows that the center of π 0 L 0 is generated by the images of the maps ϕ s , alongside with the fact that the remaining links are freely generated by some prime elements. More precisely, when Q denotes the prime string 2-links in L that do not belong to the image of a ϕ s and when Q 0 " Q X L 0 , one has the following result: Theorem 1.10 (Blair, Burke and Koytcheff, [BBK15]). The monoid π 0 L 0 is isomorphic to the product of π 0 Kˆ3 and the free (non-commutative) monoid on the basis π 0 Q 0 . Moreoever, an isomorphim is induced by the inclusion π 0 Q 0 ãÑ π 0 L 0 and the maps ϕ s .
The string links generated by the images of ϕ Ò and ϕ Ó are called split. Such a string link can also be characterized by the existence of a properly embedded disc separating the two strands in the complement. The string links generated by the image of ϕ Ù and invertible elements are called double cables. They can alternatively be defined as the links whose strands are parallel. Note that Theorem 1.10 above tells us that the center of π 0 L precisely consists of the split links, double cables and their products, and that any other string link only commutes with central elements.
We now introduce framed string 2-links. They are the 2-stranded analogue of framed long knots as they arise by thickening the two strands. Let ι : D 2 > D 2 ãÑ D 2 be the embedding that rescales the discs to make their radii 1 {8 and translates them so that they are centered at p0,˘1{2q. We refer to id Rˆι as the standard embedding pRˆD 2 q > pRˆD 2 q ãÑ RˆD 2 . z' Definition 1.11. A framed string 2-link is an embedding pRˆD 2 q>pRˆD 2 q ãÑ RˆD 2 that restricts to the standard embedding outside of pJˆD 2 q>pJˆD 2 q and maps the interior of pJˆD 2 q>pJˆD 2 q in the interior of JˆD 2 . The space of framed string 2-links is denoted EC ι p1, D 2 q. When supp ι pf q denotes the closure of tpt, xq P pRˆD 2 q > pRˆD 2 q, f pt, xq ‰ pt, ιpxqqu for any embedding f , the condition for f to lie in EC ι p1, D 2 q can be reformulated as supp ι pf q Ă pJˆD 2 q > pJˆD 2 q and f pintppJˆD 2 q > pJˆD 2 qqq Ă JˆD 2 .
Framed string 2-links again dispose of a binary concatenation operation # and a restriction map EC ι p1, D 2 q Ñ L preserving it. This endows the isotopy classes π 0 EC ι p1, D 2 q with a monoid structure with the standard embedding as unit. An obstruction for the restriction map to be an homotopy equivalence is again the framing of each strand. We define the framing number ω of an element of EC ι p1, D 2 q as in Definition 1.5, except that it now consists of a couple of integers, one for each strand.
Definition 1.12. Let f be a framed string 2-link with strands f Ò and f Ó . We define the upper framing number ω Ò pf q as the linking number lkpf Ò |Rˆp0,0q , f Ò |Rˆp0,1q q. The lower framing number ω Ó pf q is defined the same way and the whole framing number ωpf q is the couple of integers pω Ò pf q, ω Ó pf qq. The framing number is isotopy invariant and additive with respect to the concatenation for the same reasons as before. This makes it descend to a monoid morphism π 0 EC ι p1, D 2 q Ñ Zˆ2. We are now able to get rid of this twisting phenomenom by defining: Definition 1.13. The space of fat string 2-linksL is the subspace ω´1p0, 0q Ă EC ι p1, D 2 q.
Fat string 2-links are stable under concatenation. We denote byL 0 ,Q andQ 0 the elements of L whose restrictions to pRˆp0, 0qq > pRˆp0, 0qq lie in L 0 , Q and Q 0 , respectively.L is a good approximation for L in the sense that: Proposition 1.14 (Burke and Koytcheff, [BK15]). The restriction mapL Ñ L is a homotopy equivalence.
In particular, the monoid π 0L is completely understood and has the same structure as π 0 L, explicited in Theorem 1.10.
The remaining of this paper is dedicated to the elaboration of an algebraic structure on the space level of long knots and string links. The stacking products are examples of binary operations on the space level that relate to the monoids π 0 K and π 0 L. We aim to find a refinement of these operations into a more subtle structure, in order to generalize the results of Theorems 1.8 and 1.10 to the space level. These structures will formalize as operadic actions and the isomorphisms described in the theorems above will generalize as equivariant homotopy equivalences. Budney's Theorem 11 in [Bud07] precisely answers this problem in the case of knots, and Burke and Koytcheff's Theorem 6.8 in [BK15] partially deals with the case of string 2-links. In the following sections, we recall the work presented in these two papers and treat the case of string 2-links in a wider manner.

Operads
The purpose of this section is to recall the definition of (colored) operads, set up some notations and introduce the two operads of prime interest in this paper: the little cubes operad C n and a 4-colored version of the Swiss Cheese operad that we call SCL for "Swiss Cheese for Links". We also discuss the different types of algebras these objects encode and review free models for these structures.
2.1. Colored Operads and their Algebras. We start by reviewing the notion of colored operad. Let S be a set of colors. We denote by S ‹ the collection of tuples of elements of S. In other words, S ‹ is the union š ně0 Sˆn. Each Sˆn is naturally a right Σ n -space. The length of a tuple t P S ‹ is denoted |t| and, for every s P S, |t| s is the number of times s appears in t. Given two tuples of colors t, u and an integer i ď |t|, we denote by t˝i u the p|t|`|u|´1q-tuple t˝i u " pt 1 , . . . , t i´1 , u 1 , . . . , u |u| , t i`1 , . . . , t |t| q.
Note that |t˝i u| s " |t| s`| u| s whenever s ‰ t i and |t| s`| u| s´1 for s " t i . We write s n for the tuple ps, . . . , sq P Sˆn for every s P S and n ě 0. We are in the right framework to define: Definition 2.1. A colored operad O over the colors S consists of the following combined data: (i) for every t P S ‹ and s P S, a space Opt; sq, (ii) for every permutation σ P Σ n , t an n-tuple of colors and s a color, a map σ˚: Opt; sq ÝÑ Optσ; sq such that τ˚˝σ˚" pσ˝τ q˚and id˚" id, (iii) for every color s, a unit 1 s P Ops; sq, (iv) for every t, u P S ‹ and i ď |t|, operadic compositions i : Opt; sqˆOpu; t i q ÝÑ Opt˝i u; sq satisfying the usual associativity, symmetric and unital conditions. The latter are thoroughly detailed in [Yau16].
The elements of Opt, sq are called operations with inputs t and output s. The units 1 s are sometimes refered to as identites. We also often write aσ for σ˚paq and a˝i b for˝ipa, bq. A morphism of colored operads f : O Ñ P is a collection of maps f pt,sq : Opt; sq Ñ Ppt; sq preserving operadic compositions, symmetric actions and units.
The prototypical examples of colored operads are the endomorphism operads. Let X " pX s q sPS be an S-tuple of spaces. Given a vector of colors t, we note Xˆt for the product ś i X t i . Now, for any output color s, we set E X pt; sq to be the mapping space ToppXˆt, X s q. A element σ P Σ |t| can act on a f P E X pt; sq by permuting its entries, resulting in an element of E X ptσ; sq. Also, when u is another vector of colors and g lies in E X pu; t i q, one can inject g into the i th entry of f to get the composite f˝i g P E X pt˝i u; sq. This specifies the data of a colored operad E X on the colors S.
Definition 2.3. A colored operad O over a single color s is called an operad. In this case, we write Opnq for Ops n ; sq and call it the space in arity n. The unit 1 s is simply denoted 1. The operadic compositions are now maps OpnqˆOpmq Ñ Opn`m´1q and the symmetric structure turns each Opnq into a right Σ n -space.
Operads are useful to specify categories of algebraic objects. This is formalized via operadic actions which we define now.
Definition 2.4. Let O be a colored operad over the set of colors S and X " pX s q sPS an S-tuple of spaces. We say that X is a O-algebra if it comes with a morphism of operads κ : O Ñ E X . In other words, a O-algebra structure on X is a collection of maps κ pt;sq : Opt; sq ÝÑ ToppXˆt, X s q preserving the operadic compositions, identities and symmetric actions described in [Yau16]. They may also be thought of as maps Opt; sqˆXˆt Ñ X s and we shall use each framework when it is more convenient. A morphism of O-algebras f : X Ñ Y is a collection of maps f s : X s Ñ Y s preserving the operadic actions.
Example 2.5. Consider the operad obtained with the one point space in every arity and trivial symmetric actions and operadic compositions. An action of this operad on a space X is the data of a single map Xˆn Ñ X for every non-negative n. One readily checks that the required relations correspond to the associativity and commutatity of Xˆ2 Ñ X, as well as the fact that the element specified by the 0 th map Xˆ0 Ñ X acts as a unit. In other words, an action of this operad on X is a commutative topological monoid structure on X. This justifies the terminology Com for this operad.
Example 2.6. Consider the operad whose space in arity n is the discrete symmetric group Σ n as an evident right Σ n -space with the following operadic composition. For every σ P Σ n , τ P Σ m and i ď n, σ˝i τ permutes t1, . . . , n`m´1u according to σ while treating ti, . . . , i`m´1u as a single block, then shuffles the latter internally according to τ . An action of this operad on a space X is a data of a map Xˆn Ñ X for every ordering of t1, . . . , nu. One readily checks that the required relations correspond to the associativity of Xˆ2 Ñ X and the fact that the element Xˆ0 Ñ X acts as a unit. In other words, the algebras over this operad are the non-necessarily commutative topological monoids. This operad is called the associative operad and is denoted As.
2.2. Free Algebras. Before we introduce the two operads that will act on the spaces of knots and links, we take some time to discuss free algebras. When O is a colored operad with the set of colors S, the O-algebras and their morphisms form a category denoted O-Alg. There is a forgetful functor U : O-Alg Ñ TopˆS mapping a O-algebra to its underlying S-tuple of spaces. By free O-algebra, we understand the left adjoint Or s to the forgetful functor U. In other words, the free O-algebra generated by X " pX s q sPS should lead to a bijection O-AlgpOrXs, Y q -TopˆSpX, UpY qq for every O-algebra Y . A well-known model for OrXs is obtained as follows. For every vector x " px 1 , . . . , x n q P Xˆt and permutation σ, we note σx for px σ´1p1q , . . . , x σ´1pnq q P Xˆt σ´1 . We set where " identifies each pa, xq with paσ, σ´1xq for every permutation σ. The action of O is obtained by composing in the Opt; sq factor. The desired bijection above is then a formal verification. When O is an uncolored operad, " corresponds to the Σ n -orbits and we can simplify OrXs " ž n OpnqˆΣ n Xˆn.
We conclude this paragraph with a quick observation about free algebras. When O is a colored operad with set of colors S, the components π 0 O naturally inherit an operad structure. Similarly, if X " pX s q sPS is a O-algebra, then the components π 0 X " pπ 0 X s q sPS inherit a π 0 O-algebra structure. Finally, the following result will come in handy when proving that some actions yield free algebras in Section 4.
Proposition 2.7. Let X be an S-tuple of spaces. Then, π 0 pOrXsq is a model for the free algebra π 0 Orπ 0 Xs.
Proof. We have the two models Since π 0 commutes with products and coproducts, the right hand side is equal to the quotient of π 0 p š t Opt; sqˆXˆtq by the relations ra, xs " raσ, σ´1xs. The left hand side also matches this description so both spaces are the same. It is a tautological verification to see that the action of π 0 O is the same under these identifications.
2.3. The Little Cubes Operad. We introduce the little cubes operad C n and quickly discuss the algebras it encodes. It is an uncolored operad originated in [May72] in order to understand iterated loop spaces. Our treatment is very similar to the one of Budney in [Bud07].
Definition 2.8. The real functions of the form x Þ Ñ ax`b for some positive a are said to be affine increasing. A little n-cube is an application L : J n Ñ J n of the form L " l 1ˆ¨¨¨ˆln for some affine increasing functions l i . The space of k overlapping little n-cubes C n ∞ pkq is the set of configurations of k little n-cubes J n >¨¨¨> J n Ñ J n . We set C n ∞ p0q to be the one point space. Given an element L P C n ∞ pkq, we write its decomposition in little n-cubes L " À i L i . Each L i decomposes uniquely in affine increasing functions l i,1ˆ¨¨¨ˆli,n , so writing l i,j : x Þ Ñ a i,j x`b i,j gives rise to an injection C n ∞ pkq ãÑ R 2nk : L Þ Ñ pa 1,1 , b 1,1 ,¨¨¨, a k,n , b k,n q. This is used to transfer a topology on C n ∞ pkq. Considering the C ∞ -topology actually has the same outcome.
An element of C n ∞ pkq is represented by a drawing of the images of its little n-cubes. We now equip the family of spaces C n ∞ pkq with an operadic structure. Thereafter, we define the little n-cubes operad as a sub-operad by adding a disjointness conditions on the cubes.
Definition 2.9. The overlapping little n-cubes operad C n ∞ is the operad specified as follows: (i) the space in arity k is the set of configurations of little n-cubes C n ∞ pkq, (ii) For every positive integer k, l and i ď k, the operadic composition is given by Composing L P C n ∞ pkq with the one point in C n ∞ p0q discards the i th cube of L.
(iii) The action of σ P Σ l on L P C n ∞ plq permutes the little n-cubes of L, i.e Lσ " À i L σpiq .
(iv) The unit is the identity little n-cube id J n P C n ∞ p1q.
Definition 2.10. Two little n-cubes are said to be almost disjoint if the interiors of their images are disjoint. The space of k little n-cubes C n pkq is the subspace of C n ∞ pkq consisting of pairwise almost disjoint little n-cubes. We still set C n p0q to be the one point space. Operadic compositions in the overlapping little cubes operad preserve the property of being almost disjoint so the subspaces C n pkq forms a sub-operad C n called the little n-cubes operad.
We conclude this paragraph by taking a look at the operad π 0 C n . Given k little n-cubes L P C n pkq, restricting L to the center of each cube leads to an injective map t1, . . . , ku ãÑ J n , i.e an element of the configuration space conf k pJ n q. Conversely, given k points x in conf k pJ n q, one gets an element of C n pkq by considering the identical cubes centered at x whose size is the maximal size that keeps them almost disjoint. Intuitive straight line homotopies show that these two maps are homotopy inverses, so that the homotopy type of C n pkq is the one of conf k pJ n q. In particular, when n ą 1, each C n pkq is path connected so π 0 C n " Com from Example 2.5. The free algebras over this operad are the free commutative monoids. In dimension 1, the isotopy classes of C 1 pkq are indexed by the orderings of t1, . . . , nu, so π 0 C 1 " As from Example 2.6. The free As-algebras are the free non-commutative monoids.
2.4. The Operad SCL. We go through the construction of the Swiss Cheese operad for links SCL. It a 4-colored operad that is also defined in terms of little cubes. This terminology is motivated by the fact that SCL restricts to the 2-colored Swiss-Cheese operad on several pairs of colors. We then conclude by investigating the operad of components π 0 SCL. Definition 2.11. A little n-cube L " l 1ˆ¨¨¨ˆln is said to meet the lower face of the unit cube if l n p´1q "´1. Visually, this happens when the image of L intersects J n´1ˆ0 Ă J n . The configurations of k almost disjoint little n-cubes meeting the lower face of J n is denoted C n pkq and these spaces form an operad C n just as in the case of C n .
Consider the set of four colors S " to, Ò, Ó, Ùu. The notation o is meant to remind one of the "open" color in the Swiss Cheese operad as it will play a similar role. The other symbols call up to the upper and lower strands of a string link.
Definition 2.12. The Swiss Cheese operad for links SCL is specified as follow.
(i) For s P tÒ, Ó, Ùu, the only inputs t that do not lead to an empty SCLpt; sq are the monochromatic ones, t " s n . In that case, we set SCLps n ; sq " C 2 pnq. When s " o, we set SCLpt; oq " L P C 2 ∞ p|t|q, the L i s with t i " o meet the lower face of J n and are almost disjoint from any other cube, the L i s with t i " Ò are pairwise almost disjoint, same goes with Ó and Ù ( . (ii) The operadic compositions, symmetric actions and units are inherited from C 2 ∞ .
An element of SCLpt; sq is represented by a drawing of the images of its little n-cubes. We decorate the numbering of each little cube with its associated color to distinguish the cubes that simply happen to meet the lower face of J n from those that have to. The output color also appears as an index of the whole drawing. Figure 9 gives an example. It is immediate from its definition that SCL restricts to the (cubic) Swiss-Cheese operad on the pairs of colors to, Òu, to, Óu and to, Ùu. It also clearly restricts to the little cubes operad C 2 on Ò, Ó and Ù. Originally, the 2-dimensional Swiss-Cheese operad is a 2-colored operad on the set of colors to, cu. In some sense, the color o encodes an homotopy associative algebra and the color c describes part of its center. In the case of string 2-links, the center of π 0 L 0 decomposes as π 0 Kˆ3. To encode this extra information, we split the color c into three independent colors tÒ, Ó, Ùu. We now investigate π 0 SCL and its algebras. Observe that the projection map C n pkq Ñ C n´1 pkq is an homotopy equivalence for every k and n ą 1. An homotopy inverse is obtained by inflating pn´1q-cubes into n-cubes of some fixed height. This reasonning can readily be adapted to show that SCLpt; oq is homotopy equivalent to the product C 1 p|t| o qˆC 2 p|t| Ò qˆC 2 p|t| Ó qˆC 2 p|t| Ù q.
In particular, π 0 SCLpt; sq is either ‚ empty if s ‰ o and t ‰ s n , ‚ a single point if s ‰ o and t " s n , ‚ the discrete space Σ |t|o when s " o.
Therefore, an action of π 0 SCL on X " pX o , X Ò , X Ó , X Ù q is the data of a monoid structure on each space, such that the X s are commutative for s ‰ o and act on X o with compatible actions. With this description, it is easy to see with the universal property of free objects that the free such quadruplet on the basis pA, B, C, Dq is given by π 0 SCLrA, B, C, Ds " pAsrAsˆComrBsˆComrCsˆComrDs, ComrBs, ComrCs, ComrDsq, where the last three monoids act on the first one on their respective factor.

Operadic Actions
We gather here the objects introduced in the previous sections and endow the spacesK andL with operadic actions. In a first paragraph, the fat long knotsK are equipped with a C 2 -algebra structure originally exhibited by Budney in [Bud07]. In the case of fat string 2-links, there is a C 1 -algebra structure that follows from Burke and Koytcheff's work in [BK15]. We recall its construction in a second paragraph and extend it to an action of SCL in a third one.
3.1. Budney's Action on Fat Long Knots. We start with the little 2-cubes action onK originated in [Bud07]. Following Budney's work, we first define an action of the affine increasing automorphisms of R on the self-embeddings of RˆD 2 , then proceed to extend it to the 2-dimensional little cubes operad C 2 . We note CAut 1 the group of real affine increasing functions. A little 1-cube is identified with its natural extension to the real line so that C 1 p1q lives in CAut 1 . We topologize CAut 1 as we topologized C 1 p1q, which coincides with the C ∞ -topology and turns it into a topological group.
Moreover, this restricts to an action of C 1 p1q on ECp1, D 2 q that we note pL, f q Þ Ñ Lf .
Proof. That this map defines a valid action of topological group is immediate. To prove the statement about the restriction, we just need to check that Lf restricts to the identity outside of JˆD 2 , provided L P C 1 p1q and f P ECp1, D 2 q. For any t R J, L´1ptq does not lie in J because LpJq Ă J. So, for every x P D 2 , pL´1ˆid D 2 qpt, xq is outside of JˆD 2 , where f restricts to the identity. Thus Lf pt, xq " pt, xq, which proves the second part of the proposition.
(i) Given a little 2-cube L " l 1ˆl2 , we note L π for the little 1-cube l 1 . When L P C 2 pkq, L π denotes À i L i π . These little 1-cubes may overlap so L π lies in C 1 ∞ pkq but not necessarily in C 1 pkq.
(ii) For every L " l 1ˆl2 P C 2 p1q, then L t denotes the number l 2 p´1q. Again, if L P C 2 pkq, L t is the k-tuple of reals pL 1 t , . . . , L k t q P J k .
(iii) We define a partial order on the little cubes L i of an element L P C 2 pkq. This binary relation is the order generated by setting L i ă L j if and only if L i t ă L j t and the interiors of L i π and L j π intersect. Then, a permutation σ P Σ k is said to order L if the mapping i Þ Ñ L σpiq is non-decreasing. Figure 11. Illustration of the operations p q π and p q t . The permutations id, p12q and p23q order L.
We are now ready to define an action of C 2 on ECp1, D 2 q, i.e a map of operads κ : C 2 ÝÑ E ECp1,D 2 q . For every element L P C 2 pkq and permutation σ P Σ k that orders L, we set κ k pLq : ECp1, D 2 qˆk ÝÑ ECp1, D 2 q; f " pf i q i Þ ÝÑ pL σp1q π f σp1q q˝¨¨¨˝pL σpkq π f σpkq q. One can be assured that κ k pLq does not depend on σ as follows. Two choices for σ differ by a sequence of transpositions pabq such that L a and L b are incomparable, i.e such that L a π and L b π are almost disjoint. Then, supppL a π f a q and supppL b π f b q are almost disjoint as well so both orders of composition yield the same outcome. For the continuity, consider for every τ P Σ k the map κ τ k : C 2 pjqˆECp1, D 2 qˆk ÝÑ ECp1, D 2 q; pL, f q Þ ÝÑ pL τ p1q π f τ p1q q˝¨¨¨˝pL τ pkq π f τ pkq q.
Each κ τ k is continuous and coincides with κ k on F τ " tL P C 2 pkq, τ orders LuˆECp1, D 2 qˆk. The sets F τ are closed and cover C 2 pkqˆECp1, D 2 qˆk so κ k is continuous. In arity 0, we set κ 0 to be the map sending the single point in C 2 p0qˆECp1, D 2 qˆ0 to id RˆD 2 .
Even though a proof of this result is readily available in [Bud07], we provide one here as the methods and ideas at stake will be reused before the end of this section.
Proof. The operation κ 1 clearly maps the basepoint id J 2 P C 2 p1q to the identity. We need to check the compatibility of κ with the symmetric group actions and the operadic compositions. We start with the symmetric structure. Recall that a permutation τ acting on the right of L P C 2 pkq yields À i L τ piq . It also acts on the left of f " pf i q i P ECp1, D 2 qˆk to give τ f " pf τ´1piq q i . Thus, if σ is a permutation that orders Lτ , then τ˝σ orders L. This proves the needed equality We are left to prove that κ preserves operadic compositions. Given little 2-cubes L P C 2 pkq, P P C 2 plq and an integer i ď k, we need to show that κ k`l´1 pL˝i P q " κ k pLq˝i κ l pP q. Let σ and τ be permutations that respectively order L and P . Unraveling the definition of κ shows that the desired equality boils down to checking that σ˝i τ orders L˝i P . Recall the definitions of σ˝i τ and L˝i P : ‚ σ˝i τ shuffles the interval ti, . . . , i`l´1u according to τ , then permutes t1, . . . , k`l´1u according to σ while treating the shuffled interval as a single block.
‚ L˝i P is obtained from L by replacing L i with À r L i˝P r .
If L i and L j are incomparable, L i˝P r and L j also are so the result follows.
This new structure on the space of framed long knots generalizes the stacking operation in the following sense: acting with two side by side rectangles of width 1 on two knots results in their concatenation. In particular, the Com-algebra structure on π 0 ECp1, D 2 q induced from C 2 Ñ E ECp1,D 2 q is the monoid structure described in Section 1.1.
Proof. As mentioned in Section 1.1, the framing number ω descends to a morphism of monoids π 0 ECp1, D 2 q Ñ Z. Recall also that π 0 C 2 is the commutative operad Com. The integers Z form an abelian group and thus a commutative monoid. They can therefore be seen as a C 2 -algebra via the structure map C 2 Ñ Com Ñ E Z . In this framework, the framing number ω is a C 2 -algebra morphism, hence the result.
We conclude this paragraph with a quick discussion about κ. Let L be an element of C 2 pkq. The heights of the little cubes of L only appear in the formula of κ k pLq to dictate a composition order. This is done via an ordering permutation, which we defined as an element σ P Σ k such that the mapping i Þ Ñ L σpiq is non-decreasing. Here, one can replace the word "decreasing" with "increasing" and define another action with the same formula. We refer to it as Budney's reverse action. There is no substancial difference between these two versions of κ, nor is there a reason to prefer one or the other. We still introduce the two of them now because they will both play a role in the next paragraphs.

Burke and Koytcheff 's Actions on Fat
String 2-links. This paragraph is a first step towards an adaptation of Budney's work to string 2-links. Namely, we build an action of C 1 on EC ι p1, D 2 q andL. This structure has already been exhibited by Burke and Koytcheff in Theorem 6.8 of [BK15], with C 1 appearing as a sub-operad of a way bigger object called the infection operad. As before, we start with an action of CAut 1 on the embeddings pRˆD 2 q > pRˆD 2 q ãÑ RˆD 2 , then proceed to extend it to C 1 .
Proposition 3.5. The topological group CAut 1 acts on EmbppRˆD 2 q >2 , RˆD 2 q via Moreover, this restricts to an action of C 1 p1q on EC ι p1, D 2 q that we note pL, lq Þ Ñ Ll.
Proof. The proof of this is very similar to the proof of Proposition 3.1: the fact that the formula above specifies a valid action of topological group is still clear and the restriction statement is proved just as in the case of framed long knots.
To distinguish Budney's action from the one we build now, we denote the structure map by λ. The space in arity 0 still consists of a single point that λ 0 maps to id Rˆι from Definition 1.11. For any positive integer k and L P C 1 pkq, we set λ k pLq to the map that concatenates k framed string links according to the configuration of intervals L. That is, for every f " pf i q i P EC ι p1, D 2 qˆk, λ k pLqpf q : pRˆD 2 q > pRˆD 2 q ÝÑ RˆD 2 ; pt, xq Þ ÝÑ " L i f i pt, xq when t P L i pJq, pt, ιpxqq elsewhere.
The embeddings patch in a differentiable way because the little cubes are almost disjoint. The outcome lies in EC ι p1, D 2 q because supp ι pL i f i q " pL iˆi d D 2 q >2 psupp ι pf i qq. Theorem 3.6 (Burke and Koytcheff, [BK15]). The operations λ turn EC ι p1, D 2 q into a C 1 -algebra.
Proof. It is clear that λ 1 pid J q is the identity on EC ι p1, D 2 q. We check the compatibility with the symmetric actions. Let τ be a permutation, L P C 1 pkq and f P EC ι p1, D 2 qˆk. To prove the desired λ k pLτ, f q " λ k pL, τ f q, we show that these maps agree on the images of every little cube of L. This is enough as they clearly restrict to id Rˆι outside of these intervals. For every i ď k, the left hand side of the equation restricts to pLτ q i f i on pLτ q i pJ k q. The right hand side restricts to L τ piq f τ´1˝τ piq " pLτ q i f i so we are done. The associative compatibility is verified the same way.
As in the case on knots, the stacking operation arises as a special case of this new action. More precisely, acting with two side by side intervals of lenght 1 on two string links results in their concatenation. Therefore, the As-algebra structure on π 0 EC ι p1, D 2 q induced from λ : C 1 Ñ E ECιp1,D 2 q is the monoid structure on π 0 EC ι p1, D 2 q discussed in section 1.2. Moreover, as in the case of knots, we can restrict ourselves to unframed embeddings: Theorem 3.7. The fat string 2-linksL form a sub-C 1 -algebra of EC ι p1, D 2 q.
Proof. Just as in Theorem 3.4, Zˆ2 is a group that we can think of as a As-algebra and therefore a C 1 -algebra. This turns the framing number ω into a morphism of C 1 -algebras, hence the result.
3.3. The Action of SCL on Fat String 2-links. This section aims to merge the two actions defined above into a single SCL-algebra structure on the spaces of fat long knots and fat string 2links. More precisely, we build an action of SCL on the quadruplet of spaces X " pX o , X Ò , X Ó , X Ù q, where X o " EC ι p1, D 2 q and X s " ECp1, D 2 q for every s P tÒ, Ó, Ùu. We start with a lemma to ease the construction.
Lemma 3.8. There is a map Proof. The continuity of this application immediately follows the continuity of the composition in the C ∞ -topology. The purpose of this lemma is actually to check that k Ù˝l˝r k Ò > k Ó s indeed lives in EC ι p1, D 2 q. This follows from the inclusions supppk s q Ă JˆD 2 , s P tÒ, Ó, Ùu, and lpintppJˆD 2 q > pJˆD 2 qqq Ă JˆD 2 .
This map is in some way a combination of the morphisms ϕ s from Section 1.2. Indeed, if one restricts this application to the subspace tid Rˆι uˆECp1, D 2 qˆtid RˆD 2 uˆ2, the formula becomes k Ò Þ Ñ pid Rˆι q˝rk Ò > id RˆD 2 s, which is a fattened version of ϕ Ò . We denote itφ Ò . Same goes with Ó.
In the case of Ù, one is left with k Ù Þ Ñ k Ù˝p id Rˆι q. This map sends a long knot k Ù to the string link whose strands are parallel and knotted according k Ù . In other words, it is again a fattened version of ϕ Ù that we noteφ Ù .
We are now ready to define the morphism µ : SCL Ñ E X for the new action. As the values of µ pt;sq heavily depend on pt; sq, defining µ takes several steps.
We start by specifying the values of µ in monochromatic cases. The operad SCL restricts to the little cubes operad C 2 on the colors s P tÒ, Óu. We set µ to Budney's action on these colors. In other words, µ ps k ;sq " κ k from Theorem 3.3. When s " Ù, we similarly set µ pÙ k ;Ùq to Budney's reverse action, which we will still denote κ by slight abuse of notation. On the color o, SCL restricts to the operad C 2 . There is a morphism p q π : C 2 Ñ C 1 and we set µ to the composite λ˝p q π on this sub-operad.
For every s P tÒ, Ó, Ùu, the only input colors t that do not lead to an empty SCLpt; sq are the monochromatic ones, t " s k . Thus, we are left to specify µ pt;sq when s " o and the inputs are mixed. To this end, we introduce color sorting functions. Let t P S ‹ . Consider four injective (not necessarily increasing) maps α s : r|t| s s " t1, . . . , |t| s u Ñ r|t|s " t1, . . . , |t|u, s P S, whose disjoint images cover r|t|s and such that t αspiq " s for every i P r|t| s s. These maps regroup inputs of the same color and are said to sort the colors of t. Observe that each α s lifts to a map SCLpt; oq Ñ C 2 p|t| s q that discards the little cubes whose colors are different from s: The behavior of these lifts with respect to the symmetric structures on SCL and Xˆt is captured by the following relations: for every σ P Σ |t| , τ P Σ |t|s , L P SCLpt; oq and f P Xˆt, pσ˝α s qL " À i L σ˝αspiq " À i pLσq αspiq " α s pLσq, pα s˝τ qL " À i L αs˝τ piq " À i pα s Lq τ piq " pα s Lqτ, pσ˝α s qf " α s pσ´1f q, pα s˝τ qf " τ´1pα s f q.
We can finally combine the previous actions and define µ pt;oq pLq as the map where the κ on the left hand side refers to Budney's reverse action and the other two to Budney's regular action.
The continuity of µ is immediate and its values do not depend on the color sorting functions: if one chooses to replace α o with α o 1 , there is a permutation τ such that α o 1 " α o˝τ . Then, the relations above give The same argument with κ shows that the remaining α s can be replaced as well. Theorem 3.9. The operations µ turn the quadruplet X into a SCL-algebra.
Proof. First of all, it is clear that every µ ps;sq sends id J 2 to the identity. The symmetric compatibility is also quickly verified: when functions pα s q s sort the colors of t, the composites pσ´1˝α s q s sort the colors of tσ and the needed equality follows.
We are left to check the compatibility with the operadic composition. Let L P SCLpt; sq, P P SCLpu; t i q for some i. We need to show that µpL˝i P q " µpLq˝i µpP q. The validity of Budney, Burke an Koytcheff's actions (Theorems 3.3 and 3.6) conclude when t and u are monochromatic of the same color. Since there is no operation with output color s P tÒ, Ó, Ùu and input colors t ‰ s k , we are left to treat the case where s " o and t ‰ o k . When evaluated in embeddings f P Xˆt˝i u , the desired equality reads µ pt˝iu;oq pL˝i P, f q " µ pt;oq`L , f 1 , . . . , f i´1 , µ pu;t i q`P , f i , . . . , f i`|u|´1˘, f i`|u| , . . . , f |t˝iu|˘.
Assume first that t i " Ò. This forces u " Ò |u| . Let us unravel the definition of µ on both sides of the equation. Let pα s q s sort the colors of t. We may ask for α Ò p|t| Ò q " i. We sort the colors s ‰ Ò in t˝i u with functions pγ s q s mapping j to α s pjq if α s pjq ă i or to α s pjq`|u| if α s pjq ą i. The reason for this choice is γ s pL˝i P q " α s L. For the remaining γ Ò , we use the same construction on r|t| Ò´1 s and extend it to r|t˝i u| Ò s via the increasing map onto the interval ti`j, j ă |u|u. The equality reduces to We further have γ Ò pL˝i P q " α Ò L˝| t| Ò P so the validity of Budney's action (Theorem 3.3) concludes. The same manipulations treat the cases t i " Ó and Ù.
We are left to treat the case t i " o. The argument mainly consists in setting the right framework to unravel definition of µ. Let pα s q s and pβ s q s be color sorting functions for t and u, respectively. We may again ask for α o p|t| o q " i. Let pγ s q s be the color sorting functions for t˝i u one naturally builds from pα s q s and pβ s q s . More precisely, γ s agrees with α s on α s´1 ptl, l ă iuq, with α s`| u| on α s´1 ptl, l ą iuq and maps the remaining interval to the inputs s in ti`l, l ă |u|u according to β s . These choices are the ones giving γ s pL˝i P q " α s L ' pL i˝β s P q for every s P tÒ, Ó, Ùu and γ o pL˝i P q " α o L˝| s|o P . The left hand side of the equality reads K L On the other hand, we have K R . . , f γsp|t|sq q˘for every s P tÒ, Ó, Ùu.
It is easy to check from the definition of λ and Theorem 3.6 that We get the following new expression for the whole right hand side of the equation: Thus we are left to identify K factors. We previously computed K L Ò " κ`α Ò L ' pL i˝β Ò P q, γ Ò f˘. Recall that when evaluating κ, one chooses a permutation that orders α Ò L'pL i˝β Ò P q and composes the embeddings accordingly. Here, L i is a little 2-cube that meets the lower face of the unit cube. In other words, L i t "´1 and cannot get any lower. Thus, the factors pL i ' β Ò P q j f γ Ò p|t| Ò`j q can be placed in first position when computing K L Ò . This ultimately shows that The case Ó is treated the exact same way. For the case Ù, the same phenomenom with Budney's reverse action shows that the factors pL i ' β Ù P q j f γ Ù p|t| Ù`j q can be placed in last position when computing K L Ù , which again shows the desired Once again, the concatenatation comes as a special case with side by side cubes of equal width. Budney's action on knots can be recovered and one can also turn a knot into a double cable or a split link using identity cubes in SCLps; oq, s P tÒ, Ó, Ùu. More precisely, µ ps,oq pid J 2 q "φ s . This shows that the π 0 SCL-algebra structure on the quadruplet π 0 X is the data of the usual monoids π 0 ECp1, D 2 q and π 0 EC ι p1, D 2 q, together with the three distinct independant actions of π 0 ECp1, D 2 q on π 0 EC ι p1, D 2 q given by theφ s , s P tÒ, Ó, Ùu. Finally, the spaces of unframed knots and links are still stable: Theorem 3.10. The quadruplet pL 0 ,K,K,Kq forms a sub-SCL-algebra of X.
Proof. Consider the two monoids Z and Zˆ2. The first one acts on the second one in three different ways: on the first factor of Zˆ2, on the second factor or diagonally. The data of these three actions is precisely the one of a π 0 SCL-algebra structure on the quadruplet pZˆ2, Z, Z, Zq. One can think of this structure as a SCL-algebra structure. Thanks to the additive properties of the linking number with respect to the concatenation of curves, one easily checks that the framing number ω turns into a morphism of SCL-algebras. The result follows.
This action of SCL on pL 0 ,K,K,Kq combines all the structure we have met on long knots and string 2-links so far. Moreover, the isotopies exhibiting the commutativity relations discussed in Section 1.2 can all be obtained with paths in SCL from a configuration of cubes to another. The next section aims to show that this correspondance actually follows from a deeper result: an homotopy equivalence between pL 0 ,K,K,Kq and a free algebra over SCL.

Freeness Results
We prove here that the operadic actions constructed in section 3 lead to free algebras over different operads. More precisely, we first introduce the main result of Budney in [Bud07], which states thatK is homotopy equivalent as a C 2 -algebra to C 2 rPs. A second theorem proved by Burke and Koytcheff in [BK15] provides an analogous statement about the action of C 1 on a subspace ofL. We then combine these results to prove the main theorem of this paper, Theorem 4.11, stating that pL 0 ,K,K,Kq is homotopy equivalent to a free SCL-algebra. These three theorems are proved with very similar methods, most of them coming from 3-manifold topology and homotopy theory. A first paragraph recalls the concepts we need from these fields, and the following three are each decidated to a freeness theorem. The proofs of the results of Budney, Burke and Koytcheff are only quickly outlined, since thourough treatments are available in [Bud07] and [BK15]. We still dispense sketches of proofs as the arguements they involve will be useful for Theorem 4.11. 4.1. Notions of 3-dimensional Topology. We introduce some basic concepts of 3-manifold theory. The instances of 3-manifolds we will encounter mostly lie in R 3 , so they keep very nice features. Namely, they are compact, orientable connected and irreducible. It is very common when studying 3-manifolds to deal with embedded surfaces: we note S 2 the 2-sphere, D 2 the disc, A the annulus, T 2 the torus and pT 2 q #2 the 2-torus. We denote by P n the n-punctured disc, whose boundary splits as an external component B ext P n and n internal components B int P n . As for common 3-manifolds, we note B " D 2ˆJ the cylinder, homeomorphic to a 3-ball D 3 , H n " P nˆI the n-handlebody and C f Ă B the complement of a fat long knots or a fat string 2-link f . The boundary of C f is a torus when f is a fat long knot, and a 2-torus when f is a fat string 2-link. A recurring procedure in the upcomming proofs is the cutting of C f along essential surfaces. We define the latter now.
Definition 4.1. Let S be a (non-necessarily connected) orientable surface properly embedded in an orientable 3-manifold M . A disk D Ă M with BD Ă S is said to be a compressing disk for S if its boundary does not bound a disk in S. A surface that admits a compressing disk is said to be compressible, and a surface different from S 2 or D 2 admitting no compressing disk is said to be incompressible.
Definition 4.2. Let S be a bordered surface properly embedded in a 3-manifold M . A Bcompressing disk for S is a disk D whose boundary consists of two arcs α and β with α Ă S and β Ă BM , such that there is no arc γ in BS such that γ Y α bounds a disk in S. A surface that admits a B-compressing disk is said to be B-compressible. Otherwise, it is B-incompressible.  (i) S is a sphere and does not bound a ball, (ii) S is a disc whose boundary does not bound a disc in BM , (iii) S is not a sphere nor a disc, it is incompressible, B-incompressible and not B-parallel.
Spaces of embeddings of incompressible surfaces have been extensively studied by Hatcher in [Hat99]. He describes in his paper how the homotopy type of such a space depends on S. This result will be used repeatedly so we formulate a precise version here.
Theorem 4.5 (Hatcher, [Hat99]). Let M be an orientable compact connected irreducible 3-manifold and S ãÑ M an essential orientable compact surface in M . Let EmbpS, M, BSq be the space of embeddings of S in M whose values at BS are fixed. Then, the component EmbpS, M, BSq S of S ãÑ M in EmbpS, M, BSq is weakly contractible unless S is closed and the fiber of a bundle structure on M , or if S is a torus. In these exceptional cases π i EmbpS, M q " 0 for all i ą 1. In the bundle case, the inclusion of the subspace consisting of embeddings with image a fiber induces an isomorphism on π 1 . When S is a torus but not the fiber of a bundle structure, the inclusion DiffpSq ãÑ EmbpS, M q obtained by pre-composing S ãÑ M induces an isomorphism on π 1 .
Another tool of 3-manifold theory that will come in handy is the JSJ-decomposition. It provides a way to cut an irreducible manifold into simpler ones. The cut are performed along essential tori, but if one keeps on cutting a manifold until no such torus is available, the obtained decomposition might not be unique. A manifold that admits no essential torus is said to be atoroidal. In order to get a unique decomposition, one must agree not to cut the pieces that are Seifert-fibred. The latter are manifolds consisting of disjoint parallel circles forming a particular fibering. The precise definition of this fibering is looser than the notion of fiber bundle with fiber S 1 . It is specified for example in [Hat07]. The decomposition theorem we use is the following: Theorem 4.6 (Jaco, Shalen and Johannson, [JS76], [Joh79]). Every orientable, compact, irreducible 3-manifold M contains a collection of embedded, incompressible tori T so that if one removes an open tubular neighbourhood of T from M , the outcome is a disjoint union of Seifert-fibred and atoroidal manifolds. Moreover, a minimal collection of such tori is unique up to isotopy.
The minimal collection of tori T from Theorem 4.6 (or sometimes its isotopy class) is called the JSJ-decomposition of M . In the case where BM consists of a single component, the piece of the cut M containing BM is called the root of the decomposition. The tori of T bounding the root are refered to as the base-level tori of T .
Our main concern while studying an orientable compact 3-manifold M will actually be the homotopy type of the group of its boundary-fixing diffeomorphisms DiffpM, BM q. More precisely, we are intersted of the subgroup Diff d pM, BM q consisting of the diffeomorphisms whose derivatives at BM agree with those of the identity. This extra condition is relevant for our work because it enables one to post-compose a fat long knot by an element of Diff d pJˆD 2 , BpJˆD 2 qq and still end up with a fat long knot. The main brick we use to prove the three upcomming freeness theorems is the following proposition: Proof. Thanks to the stability condition, we have a well-defined post-composition map DiffpM, BM q Ñ EmbpS, M, BSq S . Restriction maps such as this one have been shown to be locally trivial, and in particular fibrations. This problem, as well as the local triviality of the restriction map EmbpM, N q Ñ EmbpM 1 , N q for a submanifold M 1 Ă M , has been treated in several articles: in [Pal60] for the case of closed manifolds and in [Cer61] for the case of bordered manifolds. A more recent exposition is provided in the third section of [KM98]: the precise result we use here is formulated as Corollary 3.7 in [KM98]. The fiber over S of this map is the subgroup of diffeomorphisms fixing S, i.e
T'he base space is weakly contractible so the inclusion of the fiber is a weak homotopy equivalence.
We are left to add the derivative condition on the diffeomorphisms.
It is proved in Kupers' book on diffeomorphism groups [Kup19] that the inclusion of the subgroup Diff d pN, BN q ãÑ DiffpN, BN q is a weak homotopy equivalence for every compact manifold N . This justifies the bottom left equivalence in the diagram of inclusions below. The right horizontal equivalences come from works of Cerf in [Cer61].
The two-out-of-three rule assures us that the top left inclusion is a weak equivalence as well. We can now conclude with this same rule in the diagram: The need to study these diffeomorphism groups arises from the following classical result in modern knot theory: Proposition 4.8. Let f be a fat long knot or a fat string 2-link. Then, the componentK f orL f of f inK orL is a model for the classifying space of Diff d pC f , BC f q. Moreover, it is a KpG, 1q.
Proof. We treat the case where f is a fat long knot, the other one is treated identically. We have the inclusion and restriction maps The map on the right hand side is a fibration thanks to Corollary 3.7 in [KM98]. Now, the diffeomorphism group Diff d pB, BBq » DiffpB, BBq is weakly contractible as proved in [Hat83]. Thus, Diff d pC f , BC f q acts properly and freely on a contractible space with quotientK l , i.eK l is a model for B Diff d pC f , BC f q. Moreover, groups of diffeomorphisms of orientable Haken bordered 3-manifolds which preserve the boundary pointwise always have vanishing higher homotopy groups by Theorem 2 in [Hat99]. Thus, the long exact sequence in homotopy coming from the fibration above assures us thatK f is a Kpπ 0 Diff d pC f , BC f q, 1q.

Budney's Freeness Theorem.
We now recall Budney's main theorem in [Bud07], which is a freeness statement about the space of fat long knots as a C 2 -algebra. Recall from Section 1.1 that the prime fat long knots are denotedP ĂK, and that they are the embeddings whose isotopy classes are prime elements in the monoid π 0K . The result we present appears as Theorem 11 in Budney's paper, and we only dispense a sketch of proof here, as the complete proof is fairly long.
Sketch of Proof. Thanks to Corollary 1.8, Proposition 2.7 and the fact that applying π 0 to the structure map κ : C 2 Ñ EK endows π 0K with its usual monoid structure, we are assured that κ induces a bijection on components. Thus, we are left to prove that it is an homotopy equivalence on each of these components.
On the component of the unknot, κ restricts to the map C 2 p0qˆPˆ0 ÑK id RˆD 2 . The complement of an unknot is a 1-handlebody H 1 , and the diffeomorphism group Diff d pH 1 , BH 1 q » DiffpH 1 , BH 1 q is contractible. Proposition 4.8 therefore gives the contractibility ofK id RˆD 2 " B Diff d pH 1 , BH 1 q so we have an equivalence in this case.
On the component of a prime knot f PP, κ restricts to the map C 2 p1qˆP f ÑP f "K f . The homotopy retracting C 2 p1q onto the identity little cube shows that this map is an equivalence as well.
Suppose now that f is a composite knot f " f 1 #¨¨¨#f n . Budney proved in [Bud06] that the base-level tori T of the JSJ-decomposition of the complement of f split C f into n`1 pieces: the complements of the prime factors C f i and a root homeomorphic to S 1ˆP n . One would like to apply Proposition 4.7 in order to split Diff d pC f , BC f q as a product of diffeomorphism groups involving each Diff d pC f i , BC f i q. But, cutting along T is not possible here as the component EmbpT 2 >¨¨¨> T 2 , C f q T is not contractible by Theorem 4.5. It is also not stable under the post-composition action of DiffpC f , BC f q. Indeed, when T i is the torus bounding C f i , two tori T i and T j can be permuted by some diffeomorphism of C f if and only if f i and f j are isotopic. Let Σ f be the subgroup of Σ n preserving the partition of t1, . . . , nu given by i " j if and only if f i and f j are isotopic. Then, by considering the components of EmbpT 2 >¨¨¨> T 2 , C f q where the image of the i th torus is isotopic to a T j with i " j, and by quotienting out these components by the parametrization of each torus, one gets a space homotopy equivalent to Σ f . It is stable under the post composition action of DiffpC f , BC f q so one can use an argument similar to the proof of Proposition 4.7 to split Diff d pC f , BC f q into a product involving each C f i . Namely, after some manipulations on the diffeomorphisms, Budney manages to split Diff d pC f , BC f q as a product for some subgroup KDiffpP n , BP n q homotopy equivalent to the pure braid group on n strands KB n . Since BKB n " conf n pJ 2 q » C 2 pnq, applying the classifying space functor B leads tô At this stage, this equivalence is merely an abstract map. But, the vast majority of the group morphisms we met are inclusion-based so Budney manages to show that this equivalence coincides with κ via explicit models. Weak equivalences are finally promoted to strong ones via an application of Whitehead's Theorem.
This result can be thought of as a generalization of Schubert's Theorem: the connected sum operation # is extended to a new algebraic structure on the space level ofK, and the isomorphism π 0K " Comrπ 0P s is extended to the C 2 -equivariant homotopy equivalenceK » C 2 rPs. Note however that Budney uses Corollary 1.8 in the very first sentence of the proof, so that this generelization is in no way an alternative argument for Schubert's result.

4.3.
Burke and Koytcheff 's Freeness Theorem. We now get to Burke and Koytcheff's result about fat string 2-links. Recall from section 1.2 thatQ 0 ĂL 0 denotes the fat string 2-links which are prime but not in the image of one of the mapsφ s , s P tÒ, Ó, Ùu. The theorem we present here is numbered 6.8 in [BK15]. We again provide a quick sketch of proof, as the ideas involved will reappear in the proof of our main result, Theorem 4.11.
Theorem 4.10 (Burke and Koytcheff, [BK15]). LetŜ 0 be the subspace ofL 0 consisting of the fat string 2-links whose prime factors lie inQ 0 . Then, the restriction of the structure map λ : C 1 rQ 0 s ÝÑŜ 0 from Theorem 3.6 is a homotopy equivalence.
Sketch of Proof. Thanks to Theorem 1.10, we are assured that λ induces a bijection on components. We are therefore again left to show that it is an equivalence of each of these components.
On the component of the trivial link, λ restricts to the map C 1 p1qˆpQ 0 qˆ0 ÑL 0 id Rˆι . The complement of id Rˆι is a 2-handlebody H 2 , and the diffeomorphism group Diff d pH 2 , BH 2 q » DiffpH 2 , BH 2 q is contractible so we can conclude as in the case of knots.
Let now f be a non-trivial element ofŜ 0 and f " f 1 #¨¨¨#f n its decomposition in non-central prime fat string 2-links. According to Theorem 4.1 in [BBK15], there are n´1 twice-punctured discs separating C f into the complements of the f i s. Moreover, as shown in the fourth step of the proof of this same theorem, these discs are unique up to isotopy. This enables us to split Diff d pC f , BC f q as the product ś i Diff d pC f i , BC f i q thanks to Proposition 4.7. Applying the classifying space functor yields the equivalenceŝ When we denote by C 1 pnq p1,...,nq the (contractible) component of C 1 pnq where the intervals appear from left to right in the order p1, . . . , nq, we can go further and writê At this stage, the equivalence is still given by an abstract map but it is easy to keep track of the identifications and see that it coincides with λ. Weak equivalences are finally promoted to strong ones via an application of Whitehead's Theorem.
Again, this result is a generalization of Theorem 1.10 as its provides a free algebraic structure on the space level ofŜ 0 , which descends to the usual free monoid on the basisQ 0 on isotopy classes. 4.4. Fat Long Knots and String 2-Links form a Free SCL-algebra. We combine in this last paragraph the theorems of Budney, Burke and Koytcheff presented above to prove a freeness result for the whole space of fat string 2-links. Namely, we prove: Theorem 4.11. The restriction of the structure map µ : SCLrQ 0 ,P,P,Ps ÝÑ pL 0 ,K,K,Kq from Theorem 3.9 is a homotopy equivalence.
Just as in the preceeding paragraphs, the proof mainly consists in reducing ourselves to each connected component and splitting the diffeomorphism group of the complement a link along suitable surfaces. We state two technical lemmas to prepare this cutting process, then proceed to the proof of the theorem.
Lemma 4.12. Let f be a fat string 2-link decomposing as f o #φ Ò pf Ò q#φ Ó pf Ó q#φ Ù pf Ù q for some element f o inŜ 0 and fat long knots f s , s P tÒ, Ó, Ùu. Then, the three vertical twice-punctured discs D cutting C f into C f o , CφÒ pf Ò q , CφÒ pf Ò q and CφÒ pf Ò q are unique up to isotopy fixing the boundary. Moreover, the component EmbpD, C f , BDq D is weakly contractible and stable under the post-composition action of DiffpC f , BC f q.
Proof. The uniqueness statement is proved in the steps 1 and 3 of Blair, Burke and Koytcheff's proof of their Theorem 4.1 in [BBK15]. Applying a diffeomorphism of DiffpC, BC f q to the punctured discs D does not change the fact that they split C f into pieces homeomorphic to the complement of f o and ϕ s pf s q, s P tÒ, Ó, Ùu. Thus, the component EmbpD, C f , BDq D is stable under the post-composition action of this diffeomorphism group. Its contractibility immediately follows from Hatcher's work on incompressible surfaces (Theorem 4.5). Figure 15. Illustration of the three discs D " D 1 > D 2 > D 3 in C f .
Lemma 4.13. Let f be a fat long knot andφ s pf q the central link obtained from f , s P tÒ, Ó, Ùu.
Consider the annulus A s in the complement Cφs pf q specified by (i) A Ò " pid Rˆι qppJˆBD 2 q > ∅q, (ii) A Ó " pid Rˆι qp∅ > pJˆBD 2 qq, Then, the isotopy class of A s is stable under the post-composition action of DiffpCφs pf q , BCφs pf q q and the component EmbpA, Cφs pf q , BAq As is weakly contractible.
A Ò A Ù Figure 16. Illustration of the annuli A Ò and A Ù .
Proof. We first treat the case where s " Ò. Consider the horizontal disc E Ă CφÒ pf q separating the two strands ofφ Ò pf q. Cutting along E yields two manifolds: an upper piece containing A Ò , homeomorphic to C f , and a lower one that is a 1-handlebody H 1 . Any two discs in CφÒ pf q sharing their boundary are isotopic because CφÒ pf q is irreducible. Therefore, given a diffeomorphism g in DiffpCφs pf q , BCφs pf q q, there is an isotopy from gpEq to E. It can be extended to a boundary preserving ambient isotopy and post-composing g by the latter shows that we may assume that gpEq " E. In other words, g preserves the cut and in particular gpA Ò q lies in the upper piece. But, A Ò is boundary parallel after the cut so it is unique up to boudary-fixing isotopy in its piece, which concludes the proof in this case. The weak contractibility statement immediately follows from Theorem 4.5. The case where s " Ó is treated the exact same way.
We now deal with the case where s " Ù. The proof of the stability statement uses the JSJdecomposition recalled in Theorem 4.6, especially the uniqueness part. The idea is to show that the JSJ-decomposition of CφÙ pf q admits a single base-level torus T Ù , which must be unique up to isotopy, and that A Ù is a suitable piece of it.
Let A Ù 1 be the annulus JˆBD 2 Ă BCφÙ pf q . The two annuli A Ù and A Ù 1 share their boundary so that their union is a torus in CφÙ pf q . Let T Ù be the torus obtained by pushing A Ù Y A Ù 1 in the interior of CφÙ pf q . It is essential and cuts CφÙ pf q into two manifolds, one containing BCφÒ pf q , A Ù and A Ù 1 that we note V , the other one homeomorphic to C f . We now proceed to show that T Ù is the base-level torus of the JSJ-decomposition of CφÙ pf q .
Claim 1. Let P 2 be a twice-punctured disc and γ Ù a curve in the interior of P 2 parallel to the external boundary circle. Then, V is homeomorphic to a 2-handlebody JˆP 2 deprived of a solid torus that is a tubular neighbourhood of 0ˆγ Ù .
Proof. This unknotting process is similar to Budney's "untwisted reembedding" described in the beginning of his paper on knot complements [Bud06]. Cutting V along A Ù results in two pieces: an external one containing T Ù and an internal one that is a (knotted) 2-handlebody H 2 . The torus T Ù is boundary parallel in the external part, so this piece is a fattened torus T 2ˆI . This shows that V is the manifold obtained by gluing a 2-handlebody H 2 and a fattened torus T 2ˆI along a specific annulus their boundary. This description also matches our new model for V so we are done.
This new model makes a lot of considerations easier since it forgets all the complexity of the knotting of f . We call the two unknotted annuli JˆB int P 2 the strands of BV . Through this identification, A Ù turns into the annulus bounding a neighbourhood of the two strands not containing T Ù , as illustrated in Figure 17. The second annulus A Ù 1 remains in the boudary as JˆBD 2 .
A Ù Figure 17. Illustration of the new model for V and A Ù .
Proof. We work with the new model for V , lying in R 3 Ă S 3 . Assume by contradiction that T is an essential torus in V . Alexander's Theorem assures us that T bounds a solid torus H 1 in S 3 . This solid torus cannot entirely lie in V for T is incompressible, so H 1 contains one of the two components BpJˆP 2 q and T Ù of BW . Actually, T being essential, one of BpJˆP 2 q and T Ù meets every meridional disc of H 1 . Since T Ù and the two strands JˆB int P 2 Ă BpJˆP 2 q are unknotted, we showed that H 1 is a companion torus of the unknot, as defined by Cromwell in the fourth chapter of [Cro04]. Therefore, it must be a trivial companion thanks to Theorem 4.2.2 in [Cro04]. In other words, T is either parallel to T Ù , or it bounds a solid torus winding once around one or both strands of BV . In all of these cases, it is not essential.
At this point, we showed that T Ù and the tori of the JSJ-decomposition of C f cut CφÒ pf q into atoroidal and Seifert-fibered pieces. Finally, T Ù cannot be removed to obtain a smaller decomposition because V can never be part of a Seifert-fibered manifold, having a boundary component homeomorphic to a 2-torus. This shows that T Ù and the tori of the JSJ-decomposition of C f form a minimal decomposition of CφÒ pf q into atoroidal and Seifert-fibered manifolds, which proves that this collection is the JSJ-decomposition of CφÒ pf q . The root V of this decomposition is bounded by BCφÙ pf q and T Ù so T Ù is the only base-level torus. Its image is therefore unique up to isotopy.
Consider now g P DiffpCφs pf q , BCφs pf q q. Thanks to the work above, there is an isotopy from gpT Ù q to T Ù . It can be extended to a boundary-fixing ambient isotopy, and post-composing g by the latter shows that we may assume that gpT Ù q " T Ù . In other words, g preserves the cut along T Ù and gpA Ù q lies in V . Let us now take a look at the isotopy classes of annuli in V . Any annulus in V whose boundary agrees with BA Ù must be unknotted. This follows from Theorem 4.2.2 in [Cro04] via an argument identical the one in the proof of Claim 2. There are therefore only two (boudary-fixing) isotopy classes of annuli in V with boundary BA Ù : the one of A Ù and the one of A Ù 1 . They cannot be permuted by a diffeomorphism of V because A Ù 1 is B-parallel and A Ù is not. This shows that g |V pA Ù q is isotopic to A Ù via an isotopy that fixes the boundary in V . This concludes this second case. The weak contractibility of the component in the embedding space again follows from Theorem 4.5.
We now implement all the tools at our disposal to complete the proof of Theorem 4.11.
Proof of Theorem 4.11. Thanks to Theorems 1.8, 1.10 and Proposition 2.7, we are assured that µ induces a bijection on components. Thus, we are left to prove that it is an homotopy equivalence on each of these components.
On the component of the unlink, µ restricts to the map SCLp∅; oqˆpL 0 ,P,P,Pqˆ∅ ÑL id Rˆι . The space SCLp∅; oq consists of a single point. The complement of the unlink is a 2-handlebody H 2 , and the diffeomorphism group Diff d pH 2 , BH 2 q » DiffpH 2 , BH 2 q is contractible. Proposition 4.8 therefore gives the contractibility ofL id Rˆι " B Diff d pH 2 , BH 2 q so we have an equivalence in this case.
On the component of a fat long knot or an element ofŜ 0 , Theorems 4.9 and 4.10 conclude because the action of SCL restricts to κ and λ on these components. where SCLpo |f |o , Ò |f | Ò , Ó |f | Ó , Ù |f | Ù ; oq p1,...,|f |oq is the component of SCLpo |f |o , Ò |f | Ò , Ó |f | Ó , Ù |f | Ù ; oq where the cubes indexed by o appear from left to right in the order p1, . . . , |f | o q and where Σ f s is the subgroup of Σ |f |s preserving the partition specified by i " j if and only if f s i is isotopic to f s j . This group acts on the cubes indexed by s in SCLpo |f |o , Ò |f | Ò , Ó |f | Ó , Ù |f | Ù ; oq p1,...,|f |oq and permutes the entries in ś iď|f |sK f s i . The inclusion SCLpo |f |o , Ò |f | Ò , Ó |f | Ó , Ù |f | Ù ; oq p1,...,|f |oq ãÑ C 2 p|f | o q p1,...,|f |oqˆśs C 2 p|f | s q is a homotopy equivalence, so that we have, by rearranging terms in the product, the natural equivalences Now, Lemma 4.13 states that the three annuli A s also satisfy the conditions of Proposition 4.7. They each split Cφs pf s q into a 2-handlebody H 2 and a manifold diffeomorphic to C f s . Actually, this second piece is precisely the image of C f s through id Rˆι when s P tÒ, Óu and C f Ù itself when s " Ù. The diffeomorphism group Diff d pH 2 , BH 2 q is contractible so we get the further natural equivalences Diff d pC f s , BC f s q ãÑ Diff d pC f s , BC f s qˆDiff d pH 2 , BH 2 q ãÑ Diff d pCφs pf s q , BCφs pf s q q.
Composing these results with Proposition 4.8 yields the natural equivalenceŝ We are now able to use Budney, Burke and Koytcheff's freeness results (Theorems 4.9 and 4.10) and our previous discussion to get the equivalence SCLrQ 0 ,P,P,Ps f » C 1 rQ 0 s f oˆź We merely have an abstract equivalence at this stage. To show that it coincides with µ, we need to check the commutativity up to homotopy of the diagram But, the spaces at stake are KpG, 1qs by Proposition 4.8 so it is enough to check the commutativity in π 1 . This verification is very similar to the end of Budney's proof of his Theorem 11 in [Bud07]. An element of π 1Lf is (an homotopy class of) a based path inL f , i.e an isotopy from f to f . The elements of π 1 B Diff d pC f , BC f q can canonically be identified with π 0 Diff d pC f , BC f q in the long exact sequence of the fibration realizingL f as B Diff d pC f , BC f q in Proposition 4.8. In this framework, picking a class φ P π 0 Diff d pC f o , BC f o q and chasing the diagram along the clockwise route turns it into an element of π 0 Diff d pC f , BC f q with its support lying between´1ˆD 2 and D 1 Ă C f , then converts it into an isotopy of f according to the construction in Proposition 4.8. Chasing φ along the counter-clockwise route converts it into an isotopy of f o in π 1Lf o , then applies µ to it. The outcome is the same as eachφ s pf s q is fixed all along this last isotopy. The same arguement shows that picking a class in π 0 Diff d pCφs pf s q , BCφs pf s q q and chasing the diagrams in both direction has the same effect.
Finally, Hatcher and McCullough showed in [HM97] that the classifying spaces of the diffeomorphism groups at stake here have the homotopy types of (aspherical finite) CW-complexes. Thus, Whitehead's Theorem promotes the weak homotopy equivalence µ to a strong one.