Over the past 10-12 years, geometric topology has entered a new era. Most of the foundational problems are solved, and there’s a fairly isolated collection of foundational problems remaining. In my mind, the two most representative ones would be the smooth 4-dimensional Poincare hypothesis, and getting a better understanding of the homotopy-type of the group of diffeomorphisms of the n-sphere (especially for n=4, but for n large as well). I want to talk about what I’d call second-order problems in low-dimensional topology, less foundational in nature and more oriented towards other goals, like relating low-dimensional topology to other areas of science. Specifically, this is an attempt to describe the “spaces of knots” subject in a way that might entice low-dimensional topologists to think about the subject.

Here is a lovely, simple theorem.

Given a non-trivial link in the 3-sphere with all pairwise linking numbers equal to zero, it is impossible to put that link into a position where every component is a round circle.

Definition: A link in S^3 is “round” if every component is the intersection of an affine-linear 2-dimensional subspace of R^4 with S^3.

The idea for the proof is that if all the components of a link are round the linking number of components would either be 0 or +-1, depending on whether or not the affine-linear 2-discs they bound in D^4 intersect or not. If the pairwise linking numbers were zero, the discs do not intersect, so shrinking the radius of the sphere produces an animation where the link component radii go to zero, and the link components remain disjoint.

A corollary of this observation is that the Borromean rings (and the Whitehead link, etc) can not be put into a position where every component is round — this holds true in R^3 as well as S^3, since stereographic projection preserves round circles.

Although the Borromean rings can not be realized by round circles in R^3, they can be realized by ellipses. Haefliger used a higher-dimensional version of the ellipsoidal Borromean rings to construct his exotic smooth embedding of S^3 in S^6, so this is an idea that “has legs.”

Here is one elliptical embedding of the Borromean rings in R^3:

x^2 + 2y^2 = 1, z=0

y^2 + 2z^2 = 1, x=0

z^2 + 2x^2 = 1, y=0

You might ask “what does all this have to do with spaces of knots?” It’s about time we got to that.

Much time has been spent in geometric topology on relatively foundational problems, like classification problems. Manifolds up to diffeomorphism. Rigid hyperbolic structures. Various cobordism relationships between manifolds, surgery relationships, and so on. These are relatively discrete-ish problems. There are times when that’s less of the case. Cerf theory, sweep outs, singularity theory, open book decompositions and Teichmuller theory all have aspects of the spaces-of-things philosophy, where one studies families.

In spaces of knots, the objects of study tend to be things like the space of all C^1-smooth embeddings S^1 –> S^3 with the C^1-metric topology. That’s the topology where one takes as a distance between two smooth embeddings f,g : S^1 –> S^3 the maximum of |f(z)-g(z)| + |f'(z)-g'(z)| where z is in the circle S^1, it is sometimes called the Whitney Topology. So in this topology two such embeddings are close only when there is a “small” isotopy from one to the other.

One of the natural reasons to study spaces of knots comes not from foundational 3-manifold theory questions, but from mechanical engineering (considered broadly!). Specifically, continuum mechanics: subjects like elasticity and plasticity. These subjects study materials and their behaviours under different stresses and conditions. The connection to spaces of knots is the idea of thinking of a physical process as a dynamical system on a state space, a space of all possible configurations. Knots are one of the most basic examples of infinite-dimensional state spaces that allow for deformable objects. A more typical continuum mechanics problem would be 2-dimensional continuua, like the study of how a plastic shopping bag deforms when its carrying groceries, or the dynamics of human flesh, or the dynamics of a big canvas tent. On the extreme end, general relativity is very close to continuum mechanics. On the more pragmatic end, high-dimensional state spaces are increasingly important in subjects like robotics where one has to plan the motion of a complex object. In that sense, spaces of knots could be viewed as a “baby” case of a much wider collection of problems.

A `physical’ dynamical systems on spaces of knots is the electrostatic potential. The idea would be to imagine a knot as being an elastic band embedded in S^3, and one places a uniform electric charge along that elastic band. The elastic band is made of rubber, so the charges do not move relative to the rubber. One can write down differential equations such as this and construct various potential functions on the space of knots Emb(S^1,S^3), see for example the work of Jun O’Hara at Tokyo Metropolitan University. Knowledge of the homotopy-type of spaces of knots tells you about what kind of critical points your potential function must have (and conversely), via traditional subjects such as Morse Theory.

Here is one of the most direct connections with low-dimensional interests. An open problem in knot theory is whether or not there is an efficient algorithm to determine if a knot is trivial, say, starting from a knot diagram. The Haken algorithm has nice implementations in Regina, but it’s exponential run-time. And although it gives one access to the isotopy to the trivial knot provided it verifies the knot is trivial, it isn’t the most convenient access one could hope for.

Consider the subspace UK of Emb(S^1,S^3) consisting of knots that are isotopic to the trivial knot. We know via Allen Hatcher’s work in the 1980’s that UK has the homotopy-type of the subspace of parametrized great circles, i.e. UK has the homotopy-type of S^3 x S^2. From this we can conclude that there exists a smooth, real-valued function UK –> R where the only critical points are the global minima, that being the great circles. At present we only know such a “potential function” exists in the weak Zermelo-Frankel sense. Due to the nature of Hatcher’s proof, we do not know the *form* of such a function. If the potential function had a nice geometric or physical interpretation (something like an electrostatic potential, for example) then perhaps the gradient-flow could be turned into an efficient mechanism to recognise trivial knots. By and large the issue of finding critical points on physically-defined potential functions Emb(S^1,S^3) –> R is an open problem. But as Hatcher shows in the final section of his paper (linked above), if you had such a potential function, you could give a new proof of the Smale Conjecture. The electrostatic potential is not the only potential function that could potentially be used in a new proof of the Smale Conjecture, the Menger curvature is another seemingly-reasonable candidate, and has its own appeal.

The person that really got the study of spaces of knots off the ground and into peoples’ imaginations is Victor Vassiliev. Vassiliev had been studying singularity theory with Arnold, in the spirit of how Arnold used singularity theory to describe the (co)homology of configuration spaces. One can think of a configuration space of points in the plane as the space of embeddings of a finite set into the plane, Emb({1,2,…,n}, R^2). That embedding space sits in the space of all maps Maps({1,2,…,n}, R^2), which is just R^{2n}. So the configuration space is the complement of a “discriminant” space, sometimes also called the “diagonal” where the points in R^2 are required to have some collisions. Similarly, the embedding space Emb(S^1, S^3) is a subspace of the mapping space Map(S^1,S^3) whose homotopy-type is known, this is S^3 x \Omega S^3. So if one is content to study (co)homology of Emb(S^1,S^3) one can study it via Spanier-Whitehead duality. This turns the relatively tricky problem of studying the (co)homology of Emb(S^1,S^3) into the somewhat more tractible problem of studying the singular maps S^1 –> S^3. The singular maps are “more tractible” precisely because they form a stratified space. You can count the double points, triple points, etc, similarly you can count the places where the derivative is zero, giving a filtration. This gives you a non-homogeneous object to work with, and suddenly there are details to study. Vassiliev went quite far with this perspective, giving a spectral sequence that converges to the (co)homology of Emb(S^1,S^n) for n at least 4. In the 3-dimensional case, it’s unclear precisely what the Vassiliev spectral sequence says about the homotopy-type of Emb(S^1,S^3), and that is an open problem. The invariants of H_0(Emb(S^1,S^3)) that it produces are known as “Vassiliev invariants” or “finite type invariants”. It remains an open problem whether or not one can distinguish knots via Vassiliev invariants. Due to the nature of their definition in terms of double points, one might expect that the key property of Vassiliev invariants is how they depend on crossing changes. You would be right!

There are some wonderful connections, though. For example, the first non-trivial finite-type invariant of knots is called “the type two invariant”. It has many interpretations, my favourite being a signed count of the number of families of “satanic circles” intersecting the knot, these are the round circles that intersect the knot in 5 points making a pentagram. See Daniel’s write-up, linked, for details. This interpretation also “has legs”. The type-2 invariant of knots, from the perspective of Vassiliev, is a cohomology class defined in H^{2n-6}(Emb(S^1,S^n)) for all n>2. So it is an isotopy invariant in dimension 3, but it is also a non-trivial cohomology class in all higher dimensions as well, having a fundamental interpretation. 2n-6 is the dimension of the first non-trivial homotopy class in Emb(S^1,S^n) that does not come from the homotopy of the free loop space on S^n. Moreover, the type-2 invariant faithfully detects this homotopy/homology class. Just as in dimension 3, it is a signed count of the number of “satanic circles” on the knot. This result appears rather tersely, here. If you want to work out the proof you’ll have to understand the relation with the long-knot space, outlined in the linked paper, first.

I’m starting to hope questions such as “do Vassiliev invariants distinguish knots” are perhaps answerable in the near future. There are a variety of ways to attack this problem but I’m increasingly drawn to a relatively formal perspective. I don’t want to bore you with too much operads verbiage, but let me tell you about the geometric-topology input to this perspective. The homotopy-type of the space of smooth embeddings Emb(S^1, S^3) has a rather beautiful description in the language of operads (operads are something like topological monoids, and are a general language for operations on spaces). The most immediate analogy I can think of would be to consider the subgroups of braid groups that preserve a system of closed curves in the punctured disc. They clearly have semi-direct product descriptions. The space of knots is comparable to that, with the key ingredient being an operad that codifies satellite operations. I call it the Splicing Operad, in reference to Larry Siebenmann’s work on JSJ-decompositions of homology 3-spheres. A key theorem that allows one to compute the homotopy-type of the splicing operad (and Emb(S^1,S^3)) is:

Given an (n+1)-component hyperbolic link L in S^3, with the components denoted (L_0,L_1,…,L_n)=L, if we know the n-component sublink (L_1,…,L_n) is the trivial link, then one can isotope L into a position in S^3 so that each of the components L_1 through L_n are round circles, and if we let G be the group of orientation-preserving isometries of S^3 that restrict to homeomorphisms of L, and which restrict to homeomorphisms also of L_0, then we can ensure that the restriction of G to S^3 \setminus L is the full group of orientation-preserving hyperbolic isometries on the exterior which preserve the L_0 cusp (and which extend to continuous functions on S^3).

So this theorem is something of a partial converse to the stated theorem at the beginning of this article. While one can’t put the Borromean rings into a position where all 3 components are round circles, one can equivariantly put the Borromean rings into a position where two of the three components are round.

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