Linking Numbers of Manifolds from First Principles

Generalizing to arbitrary dimension via the Hopf degree.

\newcommand{\kl}{\widehat{K\text{-}L}}

Carl Fredrich Gauss was an alien. Or at least, sometimes I feel that way when I look at his writings. I recently came across Gauss’ first mention of the linking number of two smooth closed curves in $\mathbb{R}^3$, which occurs offhandedly on one page of his journal, quoted below:

A principal problem at the interface of geometria situs and geometria magnitudinis will be to count the intertwinings of two closed or endless curves. Let $x, y, z$ be the coordinates of an undetermined point on the first curve; $x', y', z'$ those of a point on the second and let

$$\iint \frac{(x' - x) (dy \, dz' - dz \, dy') + (y' - y) (dz \, dx' - dx \, dz') + (z' - z) (dx \, dy' - dy \, dx')}{\left[ (x' - x)^2 + (y' - y)^2 + (z' - z)^2 \right]^{3/2}}$$

then this integral taken along both curves is

$$= 4 m \pi$$

$m$ being the number of intertwinings.

This note is Part I of my attempt to work this formula out from general principles. Here we work abstractly, thinking not about curves in $\RR^3$ but general manifolds in some higher dimensional ambient $\RR^N$. Some highlights:

• We prove
• We prove

In the case $n=m=1$ this integral becomes precisely Gauss’ expression, as we show in the followup to this post. A warning to readers: I am not a knot theorist by training, so this is likely not the most efficient route to the formula; just the one that made sense to me.

The Question

Let $M^m$ and $N^n$ be two closed smooth manifolds, embedded disjointly in $\mathbb{R}^{n+m+1}$: for example, a point and a curve in $\RR^2$, two curves in $\RR^3$ or a curve and a surface in $\RR^4$. For convenience we will often write $p=m+n+1$ for the ambient dimension. We are looking to measure something that seems pretty difficult to get ahold of at first: we want to tell if $M$ and $N$ are somehow ‘linked’ together

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or if they are not:

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A common problem in topology is that the quantity we wish to measure is not explicitly accessible from the data provided. Instead, the original situation often includes far too much irrelevant (to the question at hand) information, and we need to judiciously forget some of it to grasp what we seek. Here, there is a lot of irrelevant information about exactly how the manifolds $M$ and $N$ are embedded in $\RR^p$. For example wiggling $M$ nor $N$ a little bit without causing any intersections does not affect the information of whether they are linked, and should be discarded.

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A natural choice to formalize this might be to consider two pairs of embeddings as equivalent if they differ by a 1-parameter family of embeddings; that is, to consider maps of $M\sqcup N$ up to isotopy. But this is being too greedy: isotopy preserves too much information about our initial configuration. Specifically, it remembers not only if $M$ and $N$ are linked, but also the details of exactly how $M$ and $N$ themselves are embedded in $\RR^p$.

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To fix this we need to throw away more information - we want to stop caring about how $M$ or $N$ interacts with itself, but still remember how they interact with each other. This is a delicate balance: allowing $M$ or $N$ to self-intersect throughout a deformation suggests working with general homotopies (or perhaps homotopies through immersions) instead of homotopies through embeddings

We seek an invariant of the immersion of $M\sqcup N$ into $\RR^p$ up to homotopies where $M$ and $N$ remain disjoint from one another (though we do allow the manifolds to self-intersect throughout the homotopy).

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WARNING! THIS DELETES TOO MUCH INFORMATION! LOOK AT THE WHITEHEAD LINK. BUT BECAUSE OF THE DIFFICULTY OF KNOT THEORY, ITS OFTEN BETTER TO ERR ON THE SIDE OF THROWING TOO MUCH INFORMATION AWAY RATHER THAN TRYING TO KEEP TOO MUCH. IN FACT, THIS IS WHAT GAUSS’ NUMBER CAPTURES.

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Using the Language of Homotopy

Now we’ve precisely specified a space we are interested in, but not in a way that is very amenable to calculation. Our data consists of a pair of maps satisfying a disjointness condition, and the general machinery of topology does not have ready-built tools for such data. It does have tools for dealing with spaces of single maps however, so our first goal is to translate our data into this format. Given embeddings $K\colon M\to \RR^p$ and $L\colon N\to \RR^p$, a natural way to consider them simultaneously is to look at their product,

$K\times L\colon M\times N\mapsto \RR^p\times\RR^p$

If $K$ and $L$ are disjoint, $K(s)\neq L(t)$ for every $(s,t)\in M\times N$, and the image of $K\times L$ avoids the diagonal $\Delta = \{(x,x)\mid x\in\RR^p\}$. Thus, under the identification $(K,L)\mapsto K\times L$ disjoint homotopies correspond to homotopies of maps $M\times N\to \RR^p\times\RR^p\smallsetminus \Delta$

Precisely, this allows us to change perspective and wrap up the initially-slippery-to-describe disjointness of images into a property of the map.

This is quite nice, however there is one final technical consideration to think about. While all disjoint homotopies of $K,L$ induce a homotopy of $K\times L$ into $\RR^{2p}\smallsetminus \Delta$, not every homotopy of such a map comes from disjoint homotopies of $K,L$. This is simply because the inclusion in the above theorem is as a proper subset. As topology has much broader tools for working with spaces of maps up to any homotopy than up to some restricted notion of homotopy, we will work with this potentially¹ weaker notion of equivalence.

But this brings the quantity we want to compute in contact with objects we know how to work with, and we can define a Link Invariant

Now phrased purely in terms of homotopy, we can simplify the problem further using homotopy equivalences. The change of coordinates $(u,v)\mapsto (u,v-u)$ is a self-homeomorphism of $\RR^p$ sends the diagonal $\Delta$ to the points with zero as their second coordinate: thus,

$\RR^p\times\RR^p\smallsetminus \Delta \cong \RR^p\times\left(\RR^p\smallsetminus \{0\}\right)$

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As homotopy is insensitive to homeomorphism, we can replace our description of the codomain. This makes the overall topology of the codomain much more transparent. The first factor $\RR^p$ is contractible, and so is homotopy equivalent to a point. The second factor is a punctured $\RR^p$, which is homotopy equivalent to the $p-1$ sphere via the deformation retraction $v\mapsto \frac{v}{\|v\|}$.

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This chain of homotopy equivalences lets us replace our codomain again, yielding alltogether

$$\begin{align} [M\times N, \RR^{2p}\smallsetminus \Delta]&\simeq [M\times N, \RR^p\times(\RR^p\smallsetminus\{0\})]\\ &\simeq[M\times N, \RR^p\smallsetminus \{0\}]\\ &\simeq [M\times N, \mathbb{S}^{p-1}] \end{align}$$

It remains to track through what happens to our map $K\times L$ along this transformation, which we calculate simply by post-composing $K\times L$ with the corresponding chain of homotopy equivalences:

$$\begin{align} \left(K(s),L(t)\right)&\mapsto (K(s),K(s)-L(t))\\ &\mapsto K(s)-L(t)\\ &\mapsto \frac{K(s)-L(t)}{|K(s)-L(t)|} \end{align}$$

The final map here, sending $(s,t)$ to the normalized difference between their outputs is the natural encoding of our original disjoint embeddings $K,L$ in this simplified space. We will be using it alot, so it deserves a symbol of its own

These homotopy equivalences on the level of functions descend to bijections on the level of homotopy classes, so we can replace our original quantities defining the Link Invariant with their simplified counterparts

Linking is Picky about Dimension

With our problem phrased in terms of familiar topological objects, we can begin our work on understanding this invariant. The first major realization is a common theme in knot theory but quite surprising the first time you see it: linking is almost always trivial, and only interesting in one embedding dimension. That dimension is going to be $p=m+n+1$, or one more than the sum of the dimensions of the manifolds involved. So the interesting examples will include things like a point and curve in the plane (winding number), two closed curves in $\RR^3$ (gauss’ linking number) and a curve and surface in $\RR^4$, but not things like two surfaces in $\RR^3$ or two curves in $\RR^5$. We see this in two parts, ruling out the smaller and larger embedding dimensions by separate arguments.

First, consider the possibility the ambient space is of too small a dimension for linking to be interesting. For example, while two curves in $\RR^3$ can wind around each other in complicated ways avoiding the measure-zero space of intersections, curves in the plane cannot: when two curves intersect all nearby deformations also intersect.

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In cases like this, the interesting² data here isn’t about what we’d usually think of as linking but rather about intersections. And there is indeed some beautiful theory here, but is different than what we are trying to capture. How do we formalize this idea that ‘generic embeddings can intersect’ which generalizes to higher dimensions? One means is through the language of transversality. We say two submanifolds are transverse at a point of intersection if their tangent spaces sum to the ambient tangent space, and transverse if they are transverse at all points of intersection. Two maps are transverse if their images are transverse.

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To prove this, general theory from differential topology ensures transversality of maps is an open condition, so if two maps intersect transversely then so do all nearby deformations. Nonintersecting maps are vacuously transverse. And, if $M^n$ and $N^n$ are manifolds with $K,L$ intersecting embeddings into $\RR^p$, then $K$ and $L$ are transverse at an intersection point $u$ whenever $T_u K(M)+ T_u L(N)= T_u\RR^p$. Below we compute in terms of dimensions, using $I=T_u K(M)\cap T_u L(N)$ as a shorthand for the intersection.

$$\begin{align} \dim\left(T_u K(M)+ T_u L(N)\right) &= \dim T_u K(M) + \dim T_u L(N) - \dim I\\ &= \dim K(M)+\dim L(N) - \dim I\\ &= \dim M +\dim N - \dim I\\ &= m+n - \dim I \end{align}$$

For the sum to be $\RR^p$, this total dimension must be $p$ and so we see $m+n-\dim I =p$ Since $I$ is a vector subspace of $T_u\RR^p$ it’s dimension is $\geq 0$, and so $m+n-p=\dim I\geq 0\,\implies \, m+n\geq p$

This rules out the first half, all the way up through the case $p=m+n$. The case $p=m+n+1$ will turn out to be the interesting one, so we leave this for now, and rule out interest in larger values of $p$.

The argument here goes directly through our simplified notion of the Link Invariant, using the function space $[M\times N, \mathbb{S}^{p-1}]$. If $K$, $L$ are disjoint embeddings, then $\kl$ is a smooth map of an $m+n$ manifold into $\mathbb{S}^{p-1}$. By hypothesis, $p>m+n+1$ so the sphere’s dimension is strictly larger than that of $M\times N$, and the image under $\kl$ misses at least one point of the sphere. With a little work³ we can do this for all maps in a homotopy: that is, a homotopy of maps $M\times N\to \mathbb{S}^{p-1}$ induces a homotopy of maps $M\times N\to \RR^{p-1}$. This gives us an equivalence of homotopy classes into $\mathbb{S}^{p-1}$ with homotopy classes into $\RR^{p-1}$. But $\RR^{p-1}$ is contractible: its homotopy equivalent to a point. And all maps to a point are equal, so everything collapses:

$$\begin{align} [M\times N, \mathbb{S}^{p-1}]&\simeq [M\times N,\RR^{p-1}]\\ &\simeq [M\times N, \star]\\ &=\{\star\} \end{align}$$

Thus, the space of homotopy classes is a single point, for any $M,N$ and every pair of disjoint embeddings $K,L$ are forced to be mapped to this point by the Link Invariant: it stores no information about $K,L$ or even $M,N$.

The Linking Number as a Degree

We are left with but one interesting possibility to consider, the linking of $m$ and $n$ manifolds in $\RR^{m+n+1}$. In this Goldilocks dimension our simplified picture involves looking at maps of the $m+n$-manifold $M\times N$ into the $m+n$ sphere. Maps between manifolds and spheres of the same dimension are important across topology, and are completely classified by an famous theorem of Hopf:

A quick review of degree: a map $f\colon X\to Y$ then $f$ induces a map $f_\star H_\cdot(X,\mathbb{Z})\to H_\cdot(Y, \mathbb{Z})$. If $X,Y$ are closed and oriented then their top homologies are isomorphic to $\ZZ$, generated by the fundamental classes $[X]$, $[Y]$ respectively. So, when they have the same dimension $N$, the induced map $f_\star$ in top homology is $f_\star \colon H_N(X,\mathbb{Z})\to H_N(Y,\mathbb{Z})$

Which is essentially (using these isomorphisms) a map $\mathbb{Z}\to\mathbb{Z}$, and is fully characterized by a single integer: where the generator is sent. Thus, in top homology at least the data of $f_\star$ is fully captured by the integer $n$ such that

$f_\ast([X])= n [Y]$ We call this number the degree:

Unpacking the definition of degree gives a means of restating Hopf’s theorem that proves useful:

Footnotes

1. Homotopy theorists out there: is it actually weaker? I do not know any examples of maps $M\times N \to \RR^{2p}\smallsetminus \Delta$ which are homotopic, but are not homotopic in a factor-preserving way. ↩

2. This is not to say there are not interesting things going on with $\operatorname{Link}(K,L)$, but those interesting things relate more to homotopy properties of $M\times N$ than they do to something we’d recognize as the linking of $K(M)$ and $L(N)$. ↩

3. Our homotopies are not necessarily smooth so we might have to worry about space-filling fractal maps, but any such map can be perturbed arbitrarily small to a smooth representative of the same homotopy class. ↩