Homology of spaces of knots, known unknowns part 2: the coming-together of loose ends

Shortly after arriving as a postdoc at the University of Rochester I started getting results on the global topology of spaces of knots. This was, in a sense, the first topic I had wanted to work on for my dissertation. But I got stuck, and my Ph.D evolved to a different topic. At Rochester, sparked by Fred Cohen's curiosity, the project regained momentum.

The main result was the construction of an action of the operad of (j+1)-cubes on an embedding space I've been calling EC(j,M). This is the space of smooth embeddings of RjxM in itself, where the embeddings are required to be the identity outside of the "box", IjxM where I is the standard interval [-1,1]. M is any compact manifold. To relate this to something familiar: let K be the space of "classical" long knots. By that I mean smooth embeddings of R into R3 which agree with the inclusion t --> (t,0,0) outside of I. Then EC(1,D2) has the homotopy-type of the integers Cartesian product K. In this respect the spaces EC(j,M) could be called spaces of "framed" embeddings. The "punchline" of my results while at Rochester was that K is a free 2-cubes object. This was discussed in previous posts.

The result was satisfying but also something of a paradox to me -- that EC(1,D2) had the homotopy-type of the integers Cartesian product a free 2-cubes object didn't quite fit with some results in nearby subjects. In contrast EC(j,*) (effectively the group of diffeomorphisms of a j-cube which is the identity on the boundary) by the Cerf-Morlet "Comparison Theorem" does not appear to be close to a free object. The idealist in me wants to see more coherence in this picture (as j and M vary). The skeptic says these are relatively big objects whose homotopy-types should be rather sensitive to the ambient themes in the relevant subject areas: 3-manifolds, 4-manifolds, high-dimensional topology, and so on. Let me explain a result that is at least part-way encouraging towards a sunny, idealistic perspective.

Before I give the construction let me try to suggest a framework for how to think of the construction. When you have an action of an operad O on a space X, in some vague sense this is much like thinking of X as an analogue of a module over a ring, where the operad plays the role of ring. The cubes action on EC(j,M) is nice, but it only encodes the connect-sum operation, and there's a lot of other nice ways of constructing knots beyond the connect-sum operation. In the analogy with modules and rings, both the real and complex numbers are modules over the integers (or rationals), but this is *not* the most informative perspective -- the reals and complex numbers are isomorphic as modules over the integers (or the rationals). The philosophical problem is that the generating set is too big, or equivalently, the ring is too small to see kinds of structure we would like to see in the real and complex numbers.

Back to spaces of knots -- although long knots in R3 are "free" over the 2-cubes operad, the free generating space is too big, or equivalently the 2-cubes operad is too small -- it does not "see" enough operations on the knot space. The main operation I'm looking to encode is splicing. This is because it's closely related to geometrization, which by work of Hatcher, Ivanov and Waldhausen, means it is deeply connected to the homotopy-type of knot spaces, at least in dimension 3.

Splicing originates in the work of Schubert and his satellite operations, although Schubert did not use the word splicing. Larry Siebenmann generalized Schubert's satellite operations, and called the resulting operation splicing. Siebenmann's motivation was the JSJ-decomposition, a fundamental decomposition of 3-manifolds along embedded tori. Siebenmann was studying invariants of homology 3-spheres and was tracking how they should behave with respect to geometrization -- in particular the JSJ-decomposition. Given a homology 3-sphere, geometrization initially applies the connect-sum decomposition so one now has a finite number of prime homology 3-spheres. In these there is a (possibly empty) collection of incompressible tori that form the JSJ-decomposition, and cutting along these produces knot complements in other homology 3-spheres. The nice thing about these knot complements is they have essentially canonical geometric structures, moreover the only two geometries that occur are hyperbolic 3-dimensional geometry and the fibred Rx(hyperbolic 2-dimensional) geometry. Splicing is the reverse of geometrization: start off with some links in homology spheres and produce news links in new homology spheres in the way that "reverses" geometrization.

Here's how the construction goes. For the space EC(j,M) there will be an associated operad that I'll call the splicing operad. We'll give it the notation SC(j,M) or SC for short. An operad O consists of a family of spaces O(n) for n=0,1,2,3,... and structure maps O(n)xO(j1)x...xO(jn) --> O(j1+...+jn) satisfying some axioms. I'll first define the operad SC as a topological space, and then the maps.

If you haven't seen splicing or satellite operations before, the above picture demonstrates the idea in a nutshell, and so it will be relevant for the action of SC on EC(1,D2). In the picture there are three inputs: the Borromean rings, the trefoil and the figure-8 knot (all with one component "long"). The output is a satellite knot (bottom) with precisely two incompressible tori in the JSJ-decomposition of the complement, moreover the tori cut the knot's complement into three components: a trefoil complement, a figure-8 complement and the complement of the Borromean rings.

Going back to the ring-and-module analogy, in some sense what I'm about to describe should be considered a "ringification" of the space EC(j,M), so really I should call it an operadification of EC(j,M), for lack of a better term. I suppose an analogy would be to go from a vector space to its tensor algebra. I'm skeptical that this is the right analogy but it's something like that. Going from an algebra to some kind of universal algebra is the "right" analogy, but here the operations on EC(j,M) are satellite operations, constructed appropriately.

In the top picture, you see one element of the operad and three long knots (two above, one below). From the perspective of the (to be defined) splicing operad, one is thinking of the Borromean rings as an operation on knots. And I think this is one of the main reasons why the splicing operad is interesting. It is a collection of "operations" on knots, and yet the splicing operad consists of objects that can be interpreted as links. Moreover, since the splicing operad is an operad, it's an object that is algebraic in nature and there are many algebraic operations one can do with it. This is the "upside" of the perspective: operations and processes, things like connect-sum operations, crossing changes, satellite operations, instead of being verbs or processes, in the splicing operad they are objects and they fit into an algebraic framework. In yet another analogy: the natural numbers are a relatively simple thing to motivate (at least, compared to most other mathematical structures). In the natural numbers you have addition and multiplication, but this leads quickly to the notion of difference, and divisibility. These processes (difference, divisibility, remainders, quotients, etc) fit into a very simple mathematical formalism in the rational numbers. So what I'm saying is the splicing operad is to knots what the rational numbers are to the natural numbers: a home where operations on knots make sense in a rather simple, convenient algebraic framework.

The way to think of a splicing operation is that it consists of two rather different things. (1) It consists of a knot that is to be operated on, and (2) it consists of a way of "holding" the knot in (1) so that it can be operated on.

The objects for (1) are elements of EC(j,M) where M is any compact manifold. Usually this object is denoted L_0.

The objects for (2) are what I call "hockey pucks". They consist of an n-tuple L_1, ..., L_n, together with a permutation s, s will be an element of the symmetric group on n elements {1,2,...,n}. Each of L1,...,Ln are smooth embeddings of Ij x M into Ij x M, where I = [-1,1]. The reason they are called "hockey pucks" is because in the j=1, M=D2 case pictured above, these are embeddings of [-1,1]xD2 into itself, which when the embedding is linear it looks somewhat like a hockey puck!

The collection L=(L0,L1,...,Ln, s) are required to satisfy certain properties, and these properties will show that L is not much more than a "decorated" (n+1)-component link (at least in the case when M is a disc). But there's not much point in me writing those properties down now. The properties are formal consequences of what we want L to do, so let me write out what L is supposed to do first, and then we'll see what further properties it should have.

Given L and a collection of long knots f1, ..., fn in EC(j,M), we want L to "act" on (f1,...,fn). Moreover, the action is defined as a composite of functions, below. Let's have F=(f1,...,fn), then L.F denotes L's action on F, and it will be an element of EC(j,M). L.F is defined to be:

L.f = Ls(n).fs(n) o ... o Ls(2).fs(2) o Ls(1).fs(1) o L0

In the above, the letter "o" denotes composites of functions. The notation L_i.f_i denotes the function:

Li.fi(x) = Li(fi(Li-1(x))) provided x is in the image of Li, and

Li.fi(x) = x otherwise. So Li.fi is always function of the form Rj x M → Rj x M.

Small observation: the points where Li.fi is discontinuous are contained in the set Li(Ij x \partial M), where \partial M is the boundary of M. I call L_i(Ij x \partial M) the "annular part" of the puck Li.

The collection of all possible L's is denoted SC(n), and the union over n of all SC(n) spaces is called the splicing operad (the one that acts on EC(j,M)). So really the union of all SC(n)'s should be denoted SC(j,M) but we're running out of available notation in ASCII.

If you're new to operads, a bare-bones definition of what they are and what an action of an operad on a space is, is given here. If you're reading through those notes for the first time, other than looking at examples like the cubes operad, one rather important thing to notice is the associativity condition of an operad O acting on a space X. The associativity condition is a commutative diagram. One way around the diagram you apply the action of the operad O on the space X twice. The other way around the diagram, the action of O on X appears once, but the action of O on itself also appears. So if you're in the situation like we are (above) and you have a "candidate" operad SC, and a "candidate" action of SC on spaces EC, then one can reverse-engineer the structure maps for the operad SC from the candidate action. If you've done everything right, you can infer the structure maps for SC from this candidate action. This is exactly what I did in my paper An Operad for Splicing.

Actually, there's one more thing to observe. L's action on F is partially independant of the permutation s. So the splicing operad SC is technically defined to be a quotient object where you mod-out by that ambiguity. If two pucks are disjoint, their relative order does not matter, so consider any two such L's equivalent in that situation. More generally, even if they're not disjoint, L.F turns out to be the same provided you reorder the pucks in any way that respects the relative order (under the permutation s) of the pucks. There's another subtlety, in that the functions Li.fi have discontinuities, so we have to ensure these discontinuities do not produce discontinuities in the function L.F. This is achieved by what I call "the continuity constraint". It says that whenever s-1(i) < s-1(k) that the closure of img(Li) remove img(Lk) is disjoint from the "annular part" of Lk. This is because when we re-embed as in the proposed action, we want to ensure the new embedding avoids the annular parts of any pucks that are going to be used later in the composition to produce the embedding L.F. So that's the splicing operad. A picture of the structure maps is given, below. See the paper for details.

In dimension 3, the full splicing operad acts on EC(1,D2) in two, sort of "orthogonal" ways, in a sense it comes from the splitting of EC(1,D2) into the product of the integers Z with the space of "trivially framed" long knots, which we've been denoting K. Precisely there is a trivial bundle Z --> EC(1,D2) --> K where the map from the total space to the base space is the "restrict to Rx{(0,0)} map, which produces long knots, the fiber being the space of trivializations of the normal bundles of a fixed knot, so the fibre has the homotopy-type of the integers. A winding number construction gives a splitting EC(1,D2) --> Z, so the bundle is trivial and K can be thought of as long knots equipped with trivializations of their normal bundles, and the homological framing of the trivialization being zero (i.e. standard longitudinal). So if in a splicing operation we use as "inputs" elements from the fiber of the map EC(1,D2) --> K -- things like non-trivially framed trivial knots, then splicing produces knots where we have performed crossing changes to L0. For example, if L is the Whithead link, then the spliced knots produced are precisely twist knots. This is very interesting but it's orthogonal to my original motivation: to describe the homotopy-type of the space K. Similarly, if you splice L against trivially-framed trivial knots, you produce L_0. i.e. splicing in this sense amounts to "deleting components of L". This is again orthogonal to my interest. It's quite easy to product hyperbolic links L where one can delete a component of L to produce knots with very complicated JSJ-decompositions. So there's no hope of K being free over the operad SC -- moreover, level 0 of SC, SC(0) is EC(1,D2) on the nose! So SC doesn't even act on K.

So what I do is find a suboperad of SC that does act on K. As we've seen, we're going to want level zero of this operad to be empty, since elements of level zero erase pucks (under splicing). I asked an MO question about what to call such operads. Level O(0) of an operad is sometimes called initial so operads without O(0) could be called un-initiated or initial-less operads. That's the primary constraint on the splicing operad. The pucks L1,...,Ln are required to be orientation-preserving embeddings, and the complement of L, where L is thought of as an (n+1)-component link with L0 "long", is required to to be (1) irreducible in the sense of connect-sum operations and (2) no incompressible torus is allowed to bound a knot complement -- i.e. all incompressible tori must separate components of L. This last condition is key to the freeness result. This operad I denote SP. The main results of the paper can be summarized as:

(1) SP is an initial-less operad, moreover it's a symmetric operad, and there is a compatible action of the orhogonal group O(2). In the paper I call these (Sigma wreath O(2))-operads. The key idea is to observe that at level n of the operad SP, SP(n) there are three compatible actions: (a) one can rotate the entire element L and/or invert it. Here we're thinking of O(2) as a subgroup of SO(3), the subgroup that preserves the long axis of the knot (but not necessarily its orientation). (b) one can reparametrize the pucks L1,...,Ln and (c) one can re-label the pucks. This is an action of O(2) x (Sigma(n) wreath product O(2)).

(2) SP is a free product of (a) the semi-direct product operad of the "overlapping intervals" operad with O(2) and (b) a free operad freely generated on certain (Sigma(n) wreath O(2))-spaces. The overlapping intervals operad is equivalent to the 2-cubes operad, but the action of O(2) on the overlapping intevals operad is not the one that produces the framed discs operad. The homotopy-type of the free generating subspaces in (b) is a countable union of finite products of circles. The action of the wreath product on these spaces is linear and computable but as of yet there is no "closed form" description of the action -- but for many purposes the description is very adequate.

(3) K is free over the operad SP, and the free generating subspaces are the torus and hyperbolic knots. This completes the core story, since torus and hyperbolic knot subpsaces of K have the homotopy-type of a countable disjoint union of circles and 2-tori (S1)2.

So one of the big upshots of this is that it's a relatively compact description of the homotopy-type of long knots in R3. Describing precisely in "closed form" the homotopy-type of the generating subspaces of SP is what I've been calling "the realization problem". This boils down to either finding new bounds on the symmetries of hyperbolic links in the 3-sphere, or providing new examples (infinitely many!) of hyperbolic links with rather subtle symmetry properties. I suspect this is something that will take some time to solve. It's a very classical problem very much in the spirit of the earlier work of Makoto Sakuma's, or Akio Kawauchi's.

Looking back at the above exposition, I think a concern that hits one early might be "is this a one-off?" i.e. is there anything more to the splicing operad? Is it just a tool whose ultimate purpose is the description of knots in 3-dimensions, or is it part of a new, general perspective on embeddings, is it informative in other situations?

I suspect the answer to this question is yes, it's likely of general use in knot theory. A key observation has already been mentioned: in passing from the "efficient" splicing operad SP that acts on K to the flabby operad SC that acts on EC spaces, one gains operations such as crossing changes. This fits the splicing operad SC into the world of Skein relations relevant to certain polynomial invariants of knots. Another issue is operads have things like bar constructions so if one had some knowledge of these bar constructions, one could perhaps turn the knowledge around and talk about "derived invariants" of knots. Further, there's at least one situation where these splicing operads are known to be informative.

Victor Turchin has a paper where he develops a "Hodge Decomposition" for the homology of the space of long embeddings of R into Rn (here n>3). So this is very close to the homology of the space EC(1,Dn-1). In particular, the Hodge decomposition is believed to be induced by cabling operations on the spaces EC(1,Dn-1). Cabling is precisely what level 1 of the splicing operad SC(1,Dn-1) produces in these dimensions. So it appears that the splicing operad is producing useful homological information even for non-classical spaces of long knots. At present not enough is known about the homotopy and homology of these splicing operads but that is a natural next step. Rather simple early observations is that the components of level k of SC(1,Dn-1) are in bijective correspondence with the orbits of M(k) acting on F(k). Here F(k) is the free group on k generators, and M(k) is the McCool group. Moreover, the fundamental group of the components of SC(1,Dn-1) are essentially the stabilizers of these McCool actions. Further study of the homology and homotopy of the splicing operad seems like a rather natural direction for future research.

At higher stages of the splicing operad one gets new operations distinct from cabling -- more analogous to things like Whitehead doubling operations. In particular one gets new Browder and Dyer-Lashof type operations, which are somewhat symmetry-sensitive.

In the above picture you're seeing an element of the fundamental group of the splicing operad where you mod out by labelling and parametrization of the pucks. The motion interchanges two pucks, and switches the orientation of precisely one of the pucks. So on homology it induces a map from the Z2-stabilized part of Hi EC(1,Dn-1) to H2i+1 EC(1,Dn-1), where the Z2-action is induced by knot inversion, i.e. the action of pi0 O2 on EC(1,Dn-1). So contrary to the title of the picture it's much closer to a Dyer-Lashof type operation in form.