## Information Processing in Chemical Networks

4 January, 2017

There’s a workshop this summer:

• Dynamics, Thermodynamics and Information Processing in Chemical Networks, 13-16 June 2017, Complex Systems and Statistical Mechanics Group, University of Luxembourg. Organized by Massimiliano Esposito and Matteo Polettini.

They write, “The idea of the workshop is to bring in contact a small number of high-profile research groups working at the frontier between physics and biochemistry, with particular emphasis on the role of Chemical Networks.”

Some invited speakers include Vassily Hatzimanikatis, John Baez, Christoff Flamm, Hong Qian, Joshua D. Rabinowitz, Luca Cardelli, Erik Winfree, David Soloveichik, Stefan Schuster, David Fell and Arren Bar-Even. There will also be a session of shorter seminars by researchers from the local institutions such as Luxembourg Center for System Biomedicine. I believe attendance is by invitation only, so I’ll endeavor to make some of the ideas presented available here at this blog.

### Some of the people involved

I’m looking forward to this, in part because there will be a mix of speakers I’ve met, speakers I know but haven’t met, and speakers I don’t know yet. I feel like reminiscing a bit, and I hope you’ll forgive me these reminiscences, since if you try the links you’ll get an introduction to the interface between computation and chemical reaction networks.

In part 25 of the network theory series here, I imagined an arbitrary chemical reaction network and said:

We could try to use these reactions to build a ‘chemical computer’. But how powerful can such a computer be? I don’t know the answer.

Luca Cardelli answered my question in part 26. This was just my first introduction to the wonderful world of chemical computing. Erik Winfree has a DNA and Natural Algorithms Group at Caltech, practically next door to Riverside, and the people there do a lot of great work on this subject. David Soloveichik, now at U. T. Austin, is an alumnus of this group.

In 2014 I met all three of these folks, and many other cool people working on these theme, at a workshop I tried to summarize here:

Programming with chemical reaction networks, Azimuth, 23 March 2014.

The computational power of chemical reaction networks, 10 June 2014.

Chemical reaction network talks, 26 June 2014.

I met Matteo Polettini about a year later, at a really big workshop on chemical reaction networks run by Elisenda Feliu and Carsten Wiuf:

Trends in reaction network theory (part 1), Azimuth, 27 January 2015.

Trends in reaction network theory (part 2), Azimuth, 1 July 2015.

Polettini has his own blog, very much worth visiting. For example, you can see his view of the same workshop here:

• Matteo Polettini, Mathematical trends in reaction network theory: part 1 and part 2, Out of Equilibrium, 1 July 2015.

Finally, I met Massimiliano Esposito and Christoph Flamm recently at the Santa Fe Institute, at a workshop summarized here:

Information processing and biology, Azimuth, 7 November 2016.

So, I’ve gradually become educated in this area, and I hope that by June I’ll be ready to say something interesting about the semantics of chemical reaction networks. Blake Pollard and I are writing a paper about this now.

## Compositionality in Network Theory

29 November, 2016

I gave a talk at the workshop on compositionality at the Simons Institute for the Theory of Computing next week. I spoke about some new work with Blake Pollard. You can see the slides here:

• John Baez, Compositionality in network theory, 6 December 2016.

and a video here:

Abstract. To describe systems composed of interacting parts, scientists and engineers draw diagrams of networks: flow charts, Petri nets, electrical circuit diagrams, signal-flow graphs, chemical reaction networks, Feynman diagrams and the like. In principle all these different diagrams fit into a common framework: the mathematics of symmetric monoidal categories. This has been known for some time. However, the details are more challenging, and ultimately more rewarding, than this basic insight. Two complementary approaches are presentations of symmetric monoidal categories using generators and relations (which are more algebraic in flavor) and decorated cospan categories (which are more geometrical). In this talk we focus on the latter.

This talk assumes considerable familiarity with category theory. For a much gentler talk on the same theme, see:

## Monoidal Categories of Networks

12 November, 2016

Here are the slides of my colloquium talk at the Santa Fe Institute at 11 am on Tuesday, November 15th. I’ll explain some not-yet-published work with Blake Pollard on a monoidal category of ‘open Petri nets’:

Nature and the world of human technology are full of networks. People like to draw diagrams of networks: flow charts, electrical circuit diagrams, chemical reaction networks, signal-flow graphs, Bayesian networks, food webs, Feynman diagrams and the like. Far from mere informal tools, many of these diagrammatic languages fit into a rigorous framework: category theory. I will explain a bit of how this works and discuss some applications.

There I will be using the vaguer, less scary title ‘The mathematics of networks’. In fact, all the monoidal categories I discuss are symmetric monoidal, but I decided that too many definitions will make people unhappy.

The main new thing in this talk is my work with Blake Pollard on symmetric monoidal categories where the morphisms are ‘open Petri nets’. This allows us to describe ‘open’ chemical reactions, where chemical flow in and out. Composing these morphisms then corresponds to sticking together open Petri nets to form larger open Petri nets.

## Topological Crystals (Part 4)

28 August, 2016

Okay, let’s look at some examples of topological crystals. These are what got me excited in the first place. We’ll get some highly symmetrical crystals, often in higher-dimensional Euclidean spaces. The ‘triamond’, above, is a 3d example.

### Review

First let me remind you how it works. We start with a connected graph $X.$ This has a space $C_0(X,\mathbb{R})$ of 0-chains, which are formal linear combinations of vertices, and a space $C_1(X,\mathbb{R})$ of 1-chains, which are formal linear combinations of edges.

We choose a vertex in $X.$ Each path $\gamma$ in $X$ starting at this vertex determines a 1-chain $c_\gamma,$ namely the sum of its edges. These 1-chains give some points in $C_1(X,\mathbb{R}).$ These points are the vertices of a graph $\overline{X}$ called the maximal abelian cover of $X.$ The maximal abelian cover has an edge from $c_\gamma$ to $c_{\gamma'}$ whenever the path $\gamma'$ is obtained by adding an extra edge to $\gamma.$ We can think of this edge as a straight line segment from $c_\gamma$ to $c_{\gamma'}.$

So, we get a graph $\overline{X}$ sitting inside $C_1(X,\mathbb{R}).$ But this is a high-dimensional space. To get something nicer we’ll project down to a lower-dimensional space.

There’s boundary operator

$\partial : C_1(X,\mathbb{R}) \to C_0(X,\mathbb{R})$

sending any edge to the difference of its two endpoints. The kernel of this operator is the space of 1-cycles, $Z_1(X,\mathbb{R}).$ There’s an inner product on the space of 1-chains such that edges form an orthonormal basis, so we get an orthogonal projection

$\pi : C_1(X,\mathbb{R}) \to Z_1(X,\mathbb{R})$

We can use this to take the maximal abelian cover $\overline{X}$ and project it down to the space of 1-cycles. The hard part is checking that $\pi$ is one-to-one on $\overline{X}.$ But that’s what I explained last time! It’s true whenever our original graph $X$ has no bridges: that is, edges whose removal would disconnect our graph, like this:

So, when $X$ is a bridgeless graph, we get a copy of the maximal abelian cover embedded in $Z_1(X,\mathbb{R}).$ This is our topological crystal.

Let’s do some examples.

### Graphene

I showed you this one before, but it’s a good place to start. Let $X$ be this graph:

Since this graph has 3 edges, its space of 1-chains is 3-dimensional. Since this graph has 2 holes, its 1-cycles form a plane in that 3d space. If we take paths $\gamma$ in $X$ starting at the red vertex, form the 1-chains $c_\gamma,$ and project them down to this plane, we get this:

Here the 1-chains $c_\gamma$ are the white and red dots. They’re the vertices of the maximal abelian cover $\overline{X},$ while the line segments between them are the edges of $\overline{X}.$ Projecting these vertices and edges onto the plane of 1-cycles, we get our topological crystal:

This is the pattern of graphene, a miraculous 2-dimensional form of carbon. The more familiar 3d crystal called graphite is made of parallel layers of graphene connected with some other bonds.

Puzzle 1. Classify bridgeless connected graphs with 2 holes (or more precisely, a 2-dimensional space of 1-cycles). What are the resulting 2d topological crystals?

### Diamond

Now let’s try this graph:

Since it has 3 holes, it gives a 3d crystal:

This crystal structure is famous! It’s the pattern used by a diamond. Up to translation it has two kinds of atoms, corresponding to the two vertices of the original graph.

### Triamond

Now let’s try this graph:

Since it has 3 holes, it gives another 3d crystal:

This is also famous: it’s sometimes called a ‘triamond’. If you’re a bug crawling around on this crystal, locally you experience the same topology as if you were crawling around on a wire-frame model of a tetrahedron. But you’re actually on the maximal abelian cover!

Up to translation the triamond has 4 kinds of atoms, corresponding to the 4 vertices of the tetrahedron. Each atom has 3 equally distant neighbors lying in a plane at 120° angles from each other. These planes lie in 4 families, each parallel to one face of a regular tetrahedron. This structure was discovered by the crystallographer Laves, and it was dubbed the Laves graph by Coxeter. Later Sunada called it the ‘$\mathrm{K}_4$ lattice’ and studied its energy minimization properties. Theoretically it seems to be a stable form of carbon. Crystals in this pattern have not yet been seen, but this pattern plays a role in the structure of certain butterfly wings.

Puzzle 2. Classify bridgeless connected graphs with 3 holes (or more precisely, a 3d space of 1-cycles). What are the resulting 3d topological crystals?

### Lonsdaleite and hyperquartz

There’s a crystal form of carbon called lonsdaleite that looks like this:

It forms in meteor impacts. It does not arise as 3-dimensional topological crystal.

Puzzle 3. Show that this graph gives a 5-dimensional topological crystal which can be projected down to give lonsdaleite in 3d space:

Puzzle 4. Classify bridgeless connected graphs with 4 holes (or more precisely, a 4d space of 1-cycles). What are the resulting 4d topological crystals? A crystallographer with the wonderful name of Eon calls this one hyperquartz, because it’s a 4-dimensional analogue of quartz:

All these classification problems are quite manageable if you notice there are certain ‘boring’, easily understood ways to get new bridgeless connected graphs with $n$ holes from old ones.

### Platonic crystals

For any connected graph $X,$ there is a covering map

$q : \overline{X} \to X$

The vertices of $\overline{X}$ come in different kinds, or ‘colors’, depending on which vertex of $X$ they map to. It’s interesting to look at the group of ‘covering symmetries’, $\mathrm{Cov}(X),$ consisting of all symmetries of $\overline{X}$ that map vertices of same color to vertices of the same color. Greg Egan and I showed that if $X$ has no bridges, $\mathrm{Cov}(X)$ also acts as symmetries of the topological crystal associated to $X.$ This group fits into a short exact sequence:

$1 \longrightarrow H_1(X,\mathbb{Z}) \longrightarrow \mathrm{Cov}(X) \longrightarrow \mathrm{Aut}(X) \longrightarrow 1$

where $\mathrm{Aut}(X)$ is the group of all symmetries of $X.$ Thus, every symmetry of $X$ is covered by some symmetry of its topological crystal, while $H_1(X,\mathbb{Z})$ acts as translations of the crystal, in a way that preserves the color of every vertex.

For example consider the triamond, which comes from the tetrahedron. The symmetry group of the tetrahedron is this Coxeter group:

$\mathrm{A}_3 = \langle s_1, s_2, s_3 \;| \; (s_1s_2)^3 = (s_2s_3)^3 = s_1^2 = s_2^2 = s_3^2 = 1\rangle$

Thus, the group of covering symmetries of the triamond is an extension of $\mathrm{A}_3$ by $\mathbb{Z}^3.$ Beware the notation here: this is not the alternating group on the 3 letters. In fact it’s the permutation group on 4 letters, namely the vertices of the tetrahedron!

We can also look at other ‘Platonic crystals’. The symmetry group of the cube and octahedron is the Coxeter group

$\mathrm{B}_3 = \langle s_1, s_2, s_3 \;| \; (s_1s_2)^3 = (s_2s_3)^4 = s_1^2 = s_2^2 = s_3^2 = 1\rangle$

Since the cube has 6 faces, the graph formed by its vertices and edges a 5d space of 1-cycles. The corresponding topological crystal is thus 5-dimensional, and its group of covering symmetries is an extension of $\mathrm{B}_3$ by $\mathbb{Z}^5.$ Similarly, the octahedron gives a 7-dimensional topological crystal whose group of covering symmetries, an extension of $\mathrm{B}_3$ by $\mathbb{Z}^7.$

The symmetry group of the dodecahedron and icosahedron is

$\mathrm{H}_3 = \langle s_1, s_2, s_3 \;| \; (s_1s_2)^3 = (s_2s_3)^5= s_1^2 = s_2^2 = s_3^2 = 1\rangle$

and these solids give crystals of dimensions 11 and 19. If you’re a bug crawling around on the the second of these, locally you experience the same topology as if you were crawling around on a wire-frame model of a icosahedron. But you’re actually in 19-dimensional space, crawling around on the maximal abelian cover!

There is also an infinite family of degenerate Platonic solids called ‘hosohedra’ with two vertices, $n$ edges and $n$ faces. These faces cannot be made flat, since each face has just 2 edges, but that is not relevant to our construction: the vertices and edges still give a graph. For example, when $n = 6,$ we have the ‘hexagonal hosohedron’:

The corresponding crystal has dimension $n-1,$ and its group of covering symmetries is an extension of $\mathrm{S}_n \times \mathbb{Z}/2$ by $\mathbb{Z}^{n-1}.$ The case $n = 3$ gives the graphene crystal, while $n = 4$ gives the diamond.

### Exotic crystals

We can also get crystals from more exotic highly symmetrical graphs. For example, take the Petersen graph:

Its symmetry group is the symmetric group $\mathrm{S}_5.$ It has 10 vertices and 15 edges, so its Euler characteristic is $-5,$ which implies that its space of 1-cycles is 6-dimensional. It thus gives a 6-dimensional crystal whose group of covering symmetries is an extension of $\mathrm{S}_5$ by $\mathbb{Z}^6.$

Two more nice examples come from Klein’s quartic curve, a Riemann surface of genus three on which the 336-element group $\mathrm{PGL}(2,\mathbb{F}_7)$ acts as isometries. These isometries preserve a tiling of Klein’s quartic curve by 56 triangles, with 7 meeting at each vertex. This picture is topologically correct, though not geometrically:

From this tiling we obtain a graph $X$ embedded in Klein’s quartic curve. This graph has $56 \times 3 / 2 = 84$ edges and $56 \times 3 / 7 = 24$ vertices, so it has Euler characteristic $-60.$ It thus gives a 61-dimensional topological crystal whose group of covering symmetries is extension of $\mathrm{PGL}(2,\mathbb{F}_7)$ by $\mathbb{Z}^{61}.$

There is also a dual tiling of Klein’s curve by 24 heptagons, 3 meeting at each vertex. This gives a graph with 84 edges and 56 vertices, hence Euler characteristic $-28.$ From this we obtain a 29-dimensional topological crystal whose group of covering symmetries is an extension of $\mathrm{PGL}(2,\mathbb{F}_7)$ by $\mathbb{Z}^{29}.$

### The packing fraction

Now that we’ve got a supply of highly symmetrical crystals in higher dimensions, we can try to study their structure. We’ve only made a bit of progress on this.

One interesting property of a topological crystal is its ‘packing fraction’. I like to call the vertices of a topological crystal atoms, for the obvious reason. The set $A$ of atoms is periodic. It’s usually not a lattice. But it’s contained in the lattice $L$ obtained by projecting the integral 1-chains down to the space of 1-cycles:

$L = \{ \pi(c) : \; c \in C_1(X,\mathbb{Z}) \}$

We can ask what fraction of the points in this lattice are actually atoms. Let’s call this the packing fraction. Since $Z_1(X,\mathbb{Z})$ acts as translations on both $A$ and $L,$ we can define it to be

$\displaystyle{ \frac{|A/Z_1(X,\mathbb{Z})|}{|L/Z_1(X,\mathbb{Z})|} }$

For example, suppose $X$ is the graph that gives graphene:

Then the packing fraction is 2/3, as can be seen here:

For any bridgeless connected graph $X,$ it turns out that the packing fraction equals

$\displaystyle{ \frac{|V|}{|T|} }$

where $V$ is the set of vertices and $T$ is the set of spanning trees. The main tool used to prove this is Bacher, de la Harpe and Nagnibeda’s work on integral cycles and integral cuts, which in turn relies on Kirchhoff’s matrix tree theorem.

Greg Egan used Mathematica to count the spanning trees in the examples discussed above, and this let us work out their packing fractions. They tend to be very low! For example, the maximal abelian cover of the dodecahedron gives an 11-dimensional crystal with packing fraction 1/27,648, while the heptagonal tiling of Klein’s quartic gives a 29-dimensional crystal with packing fraction 1/688,257,368,064,000,000.

So, we have some very delicate, wispy crystals in high-dimensional spaces, built from two simple ideas in topology: the maximal abelian cover of a graph, and the natural inner product on 1-chains. They have intriguing connections to tropical geometry, but they are just beginning to be understood in detail. Have fun with them!

For more, see:

• John Baez, Topological crystals.

Part 1 – the basic idea.

Part 2 – the maximal abelian cover of a graph.

Part 3 – constructing topological crystals.

Part 4 – examples of topological crystals.

## Topological Crystals (Part 3)

6 August, 2016

Last time I explained how to create the ‘maximal abelian cover’ of a connected graph. Now I’ll say more about a systematic procedure for embedding this into a vector space. That will give us a topological crystal, like the one above.

Some remarkably symmetrical patterns arise this way! For example, starting from this graph:

we get this:

Nature uses this pattern for crystals of graphene.

Starting from this graph:

we get this:

Nature uses this for crystals of diamond! Since the construction depends only on the topology of the graph we start with, we call this embedded copy of its maximal abelian cover a topological crystal.

Today I’ll remind you how this construction works. I’ll also outline a proof that it gives an embedding of the maximal abelian cover if and only if the graph has no bridges: that is, edges that disconnect the graph when removed. I’ll skip all the hard steps of the proof, but they can be found here:

• John Baez, Topological crystals.

### The homology of graphs

I’ll start with some standard stuff that’s good to know. Let $X$ be a graph. Remember from last time that we’re working in a setup where every edge $e$ goes from a vertex called its source $s(e)$ to a vertex called its target $t(e)$. We write $e: x \to y$ to indicate that $e$ is going from $x$ to $y$. You can think of the edge as having an arrow on it, and if you turn the arrow around you get the inverse edge, $e^{-1}: y \to x$. Also, $e^{-1} \ne e$.

The group of integral 0-chains on $X$, $C_0(X,\mathbb{Z})$, is the free abelian group on the set of vertices of $X$. The group of integral 1-chains on $X$, $C_1(X,\mathbb{Z})$, is the quotient of the free abelian group on the set of edges of $X$ by relations $e^{-1} = -e$ for every edge $e$. The boundary map is the homomorphism

$\partial : C_1(X,\mathbb{Z}) \to C_0(X,\mathbb{Z})$

such that

$\partial e = t(e) - s(e)$

for each edge $e$, and

$Z_1(X,\mathbb{Z}) = \ker \partial$

is the group of integral 1-cycles on $X$.

Remember, a path in a graph is a sequence of edges, the target of each one being the source of the next. Any path $\gamma = e_1 \cdots e_n$ in $X$ determines an integral 1-chain:

$c_\gamma = e_1 + \cdots + e_n$

For any path $\gamma$ we have

$c_{\gamma^{-1}} = -c_{\gamma},$

and if $\gamma$ and $\delta$ are composable then

$c_{\gamma \delta} = c_\gamma + c_\delta$

Last time I explained what it means for two paths to be ‘homologous’. Here’s the quick way to say it. There’s groupoid called the fundamental groupoid of $X$, where the objects are the vertices of $X$ and the morphisms are freely generated by the edges except for relations saying that the inverse of $e: x \to y$ really is $e^{-1}: y \to x$. We can abelianize the fundamental groupoid by imposing relations saying that $\gamma \delta = \delta \gamma$ whenever this equation makes sense. Each path $\gamma : x \to y$ gives a morphism which I’ll call $[[\gamma]] : x \to y$ in the abelianized fundamental groupoid. We say two paths $\gamma, \gamma' : x \to y$ are homologous if $[[\gamma]] = [[\gamma']]$.

Here’s a nice thing:

Lemma A. Let $X$ be a graph. Two paths $\gamma, \delta : x \to y$ in $X$ are homologous if and only if they give the same 1-chain: $c_\gamma = c_\delta$.

Proof. See the paper. You could say they give ‘homologous’ 1-chains, too, but for graphs that’s the same as being equal.   █

We define vector spaces of 0-chains and 1-chains by

$C_0(X,\mathbb{R}) = C_0(X,\mathbb{Z}) \otimes \mathbb{R}, \qquad C_1(X,\mathbb{R}) = C_1(X,\mathbb{Z}) \otimes \mathbb{R},$

respectively. We extend the boundary map to a linear map

$\partial : C_1(X,\mathbb{R}) \to C_0(X,\mathbb{R})$

We let $Z_1(X,\mathbb{R})$ be the kernel of this linear map, or equivalently,

$Z_1(X,\mathbb{R}) = Z_0(X,\mathbb{Z}) \otimes \mathbb{R} ,$

and we call elements of this vector space 1-cycles. Since $Z_1(X,\mathbb{Z})$ is a free abelian group, it forms a lattice in the space of 1-cycles. Any edge of $X$ can be seen as a 1-chain, and there is a unique inner product on $C_1(X,\mathbb{R})$ such that edges form an orthonormal basis (with each edge $e^{-1}$ counting as the negative of $e$.) There is thus an orthogonal projection

$\pi : C_1(X,\mathbb{R}) \to Z_1(X,\mathbb{R}) .$

This is the key to building topological crystals!

### The embedding of atoms

We now come to the main construction, first introduced by Kotani and Sunada. To build a topological crystal, we start with a connected graph $X$ with a chosen basepoint $x_0$. We define an atom to be a homology class of paths starting at the basepoint, like

$[[\alpha]] : x_0 \to x$

Last time I showed that these atoms are the vertices of the maximal abelian cover of $X$. Now let’s embed these atoms in a vector space!

Definition. Let $X$ be a connected graph with a chosen basepoint. Let $A$ be its set of atoms. Define the map

$i : A \to Z_1(X,\mathbb{R})$

by

$i([[ \alpha ]]) = \pi(c_\alpha) .$

That $i$ is well-defined follows from Lemma A. The interesting part is this:

Theorem A. The following are equivalent:

(1) The graph $X$ has no bridges.

(2) The map $i : A \to Z_1(X,\mathbb{R})$ is one-to-one.

Proof. The map $i$ is one-to-one if and only if for any atoms $[[ \alpha ]]$ and $[[ \beta ]]$, $i([[ \alpha ]]) = i([[ \beta ]])$ implies $[[ \alpha ]]= [[ \beta ]]$. Note that $\gamma = \beta^{-1} \alpha$ is a path in $X$ with $c_\gamma = c_{\alpha} - c_\beta$, so

$\pi(c_\gamma) = \pi(c_{\alpha} - c_\beta) = i([[ \alpha ]]) - i([[ \beta ]])$

Since $\pi(c_\gamma)$ vanishes if and only if $c_\gamma$ is orthogonal to every 1-cycle, we have

$c_{\gamma} \textrm{ is orthogonal to every 1-cycle} \; \iff \; i([[ \alpha ]]) = i([[ \beta ]])$

On the other hand, Lemma A says

$c_\gamma = 0 \; \iff \; [[ \alpha ]]= [[ \beta ]].$

Thus, to prove (1)$\iff$(2), it suffices to that show that $X$ has no bridges if and only if every 1-chain $c_\gamma$ orthogonal to every 1-cycle has $c_\gamma =0$. This is Lemma D below.   █

The following lemmas are the key to the theorem above — and also a deeper one saying that if $X$ has no bridges, we can extend $i : A \to Z_1(X,\mathbb{R})$ to an embedding of the whole maximal abelian cover of $X$.

For now, we just need to show that any nonzero 1-chain coming from a path in a bridgeless graph has nonzero inner product with some 1-cycle. The following lemmas, inspired by an idea of Ilya Bogdanov, yield an algorithm for actually constructing such a 1-cycle. This 1-cycle also has other desirable properties, which will come in handy later.

To state these, let a simple path be one in which each vertex appears at most once. Let a simple loop be a loop $\gamma : x \to x$ in which each vertex except $x$ appears at most once, while $x$ appears exactly twice, as the starting point and ending point. Let the support of a 1-chain $c$, denoted $\mathrm{supp}(c)$, be the set of edges $e$ such that $\langle c, e\rangle> 0$. This excludes edges with $\langle c, e \rangle= 0$, but also those with $\langle c , e \rangle < 0$, which are inverses of edges in the support. Note that

$c = \sum_{e \in \mathrm{supp}(c)} \langle c, e \rangle .$

Thus, $\mathrm{supp}(c)$ is the smallest set of edges such that $c$ can be written as a positive linear combination of edges in this set.

Okay, here are the lemmas!

Lemma B. Let $X$ be any graph and let $c$ be an integral 1-cycle on $X$. Then for some $n$ we can write

$c = c_{\sigma_1} + \cdots + c_{\sigma_n}$

where $\sigma_i$ are simple loops with $\mathrm{supp}(c_{\sigma_i}) \subseteq \mathrm{supp}(c)$.

Proof. See the paper. The proof is an algorithm that builds a simple loop $\sigma_1$ with$\mathrm{supp}(c_{\sigma_1}) \subseteq \mathrm{supp}(c)$. We subtract this from $c$, and if the result isn’t zero we repeat the algorithm, continuing to subtract off 1-cycles $c_{\sigma_i}$ until there’s nothing left.   █

Lemma C. Let $\gamma: x \to y$ be a path in a graph $X$. Then for some $n \ge 0$ we can write

$c_\gamma = c_\delta + c_{\sigma_1} + \cdots + c_{\sigma_n}$

where $\delta: x \to y$ is a simple path and $\sigma_i$ are simple loops with $\mathrm{supp}(c_\delta), \mathrm{supp}(c_{\sigma_i}) \subseteq \mathrm{supp}(c_\gamma)$.

Proof. This relies on the previous lemma, and the proof is similar — but when we can’t subtract off any more $c_{\sigma_i}$’s we show what’s left is $c_\delta$ for a simple path $\delta: x \to y$.   █

Lemma D. Let $X$ be a graph. Then the following are equivalent:

(1) $X$ has no bridges.

(2) For any path $\gamma$ in $X$, if $c_\gamma$ is orthogonal to every 1-cycle then $c_\gamma = 0$.

Proof. It’s easy to show a bridge $e$ gives a nonzero 1-chain $c_e$ that’s orthogonal to all 1-cycles, so the hard part is showing that for a bridgeless graph, if $c_\gamma$ is orthogonal to every 1-cycle then $c_\gamma = 0$. The idea is to start with a path for which $c_\gamma \ne 0$. We hit this path with Lemma C, which lets us replace $\gamma$ by a simple path $\delta$. The point is that a simple path is a lot easier to deal with than a general path: a general path could wind around crazily, passing over every edge of our graph multiple times.

Then, assuming $X$ has no bridges, we use Ilya Bogdanov’s idea to build a 1-cycle that’s not orthogonal to $c_\delta$. The basic idea is to take the path $\delta : x \to y$ and write it out as $\delta = e_1 \cdots e_n$. Since the last edge $e_n$ is not a bridge, there must be a path from $y$ back to $x$ that does not use the edge $e_n$ or its inverse. Combining this path with $\delta$ we can construct a loop, which gives a cycle having nonzero inner product with $c_\delta$ and thus with $c_\gamma$.

I’m deliberately glossing over some difficulties that can arise, so see the paper for details!   █

### Embedding the whole crystal

Okay: so far, we’ve taken a connected bridgeless graph $X$ and embedded its atoms into the space of 1-cycles via a map

$i : A \to Z_1(X,\mathbb{R}) .$

These atoms are the vertices of the maximal abelian cover $\overline{X}$. Now we’ll extend $i$ to an embedding of the whole graph $\overline{X}$ — or to be precise, its geometric realization $|\overline{X}|$. Remember, for us a graph is an abstract combinatorial gadget; its geometric realization is a topological space where the edges become closed intervals.

The idea is that just as $i$ maps each atom to a point in the vector space $Z_1(X,\mathbb{R})$, $j$ maps each edge of $|\overline{X}|$ to a straight line segment between such points. These line segments serve as the ‘bonds’ of a topological crystal. The only challenge is to show that these bonds do not cross each other.

Theorem B. If $X$ is a connected graph with basepoint, the map $i : A \to Z_1(X,\mathbb{R})$ extends to a continuous map

$j : |\overline{X}| \to Z_1(X,\mathbb{R})$

sending each edge of $|\overline{X}|$ to a straight line segment in $Z_1(X,\mathbb{R})$. If $X$ has no bridges, then $j$ is one-to-one.

Proof. The first part is easy; the second part takes real work! The problem is to show the edges don’t cross. Greg Egan and I couldn’t do it using just Lemma D above. However, there’s a nice argument that goes back and uses Lemma C — read the paper for details.

As usual, history is different than what you read in math papers: David Speyer gave us a nice proof of Lemma D, and that was good enough to prove that atoms are mapped into the space of 1-cycles in a one-to-one way, but we only came up with Lemma C after weeks of struggling to prove the edges don’t cross.   █

### Connections to tropical geometry

Tropical geometry sets up a nice analogy between Riemann surfaces and graphs. The Abel–Jacobi map embeds any Riemann surface $\Sigma$ in its Jacobian, which is the torus $H_1(\Sigma,\mathbb{R})/H_1(\Sigma,\mathbb{Z})$. We can similarly define the Jacobian of a graph $X$ to be $H_1(X,\mathbb{R})/H_1(X,\mathbb{Z})$. Theorem B yields a way to embed a graph, or more precisely its geometric realization $|X|$, into its Jacobian. This is the analogue, for graphs, of the Abel–Jacobi map.

After I put this paper on the arXiv, I got an email from Matt Baker saying that he had already proved Theorem A — or to be precise, something that’s clearly equivalent. It’s Theorem 1.8 here:

• Matthew Baker and Serguei Norine, Riemann–Roch and Abel–Jacobi theory on a finite graph.

This says that the vertices of a bridgeless graph $X$ are embedded in its Jacobian by means of the graph-theoretic analogue of the Abel–Jacobi map.

What I really want to know is whether someone’s written up a proof that this map embeds the whole graph, not just its vertices, into its Jacobian in a one-to-one way. That would imply Theorem B. For more on this, try my conversation with David Speyer.

Anyway, there’s a nice connection between topological crystallography and tropical geometry, and not enough communication between the two communities. Once I figure out what the tropical folks have proved, I will revise my paper to take that into account.

Next time I’ll talk about more examples of topological crystals!

Part 1 – the basic idea.

Part 2 – the maximal abelian cover of a graph.

Part 3 – constructing topological crystals.

Part 4 – examples of topological crystals.

## Topological Crystals (Part 2)

27 July, 2016

We’re building crystals, like diamonds, purely from topology. Last time I said how: you take a graph $X$ and embed its maximal abelian cover into the vector space $H_1(X,\mathbb{R}).$ Now let me say a bit more about the maximal abelian cover. It’s not nearly as famous as the universal cover, but it’s very nice.

First I’ll whiz though the basic idea, and then I’ll give the details.

### The basic idea

By ‘space’ let me mean a connected topological space that’s locally nice. The basic idea is that if $X$ is some space, its universal cover $\widetilde{X}$ is a covering space of $X$ that covers all other covering spaces of $X.$ The maximal abelian cover $\overline{X}$ has a similar universal property—but it’s abelian, and it covers all abelian connected covers. A cover is abelian if its group of deck transformations is abelian.

The cool part is that universal covers are to homotopy theory as maximal abelian covers are to homology theory.

What do I mean by that? For starters, points in $\widetilde{X}$ are just homotopy classes of paths in $X$ starting at some chosen basepoint. And the points in $\overline{X}$ are just ‘homology classes’ of paths starting at the basepoint.

But people don’t talk so much about ‘homology classes’ of paths. So what do I mean by that? Here a bit of category theory comes in handy. Homotopy classes of paths in $X$ are morphisms in the fundamental groupoid of $X.$ Homology classes of paths are morphisms in the abelianized version of the fundamental groupoid!

But wait a minute — what does that mean? Well, we can abelianize any groupoid by imposing the relations

$f g = g f$

whenever it makes sense to do so. It makes sense to do so when you can compose the morphisms $f : x \to y$ and $g : x' \to y'$ in either order, and the resulting morphisms $f g$ and $g f$ have the same source and the same target. And if you work out what that means, you’ll see it means

$x = y = x' = y'$

But now let me say it all much more slowly, for people who want a more relaxed treatment.

### The details

There are lots of slightly different things called ‘graphs’ in mathematics, but in topological crystallography it’s convenient to work with one that you’ve probably never seen before. This kind of graph has two copies of each edge, one pointing in each direction.

So, we’ll say a graph $X = (E,V,s,t,i)$ has a set $V$ of vertices, a set $E$ of edges, maps $s,t : E \to V$ assigning to each edge its source and target, and a map $i : E \to E$ sending each edge to its inverse, obeying

$s(i(e)) = t(e), \quad t(i(e)) = s(e) , \qquad i(i(e)) = e$

and

$i(e) \ne e$

for all $e \in E.$

That inequality at the end will make category theorists gag: definitions should say what’s true, not what’s not true. But category theorists should be able to see what’s really going on here, so I leave that as a puzzle.

For ordinary folks, let me repeat the definition using more words. If $s(e) = v$ and $t(e) = w$ we write $e : v \to w,$ and draw $e$ as an interval with an arrow on it pointing from $v$ to $w.$ We write $i(e)$ as $e^{-1},$ and draw $e^{-1}$ as the same interval as $e,$ but with its arrow reversed. The equations obeyed by $i$ say that taking the inverse of $e : v \to w$ gives an edge $e^{-1} : w \to v$ and that $(e^{-1})^{-1} = e.$ No edge can be its own inverse.

A map of graphs, say $f : X \to X',$ is a pair of functions, one sending vertices to vertices and one sending edges to edges, that preserve the source, target and inverse maps. By abuse of notation we call both of these functions $f.$

I started out talking about topology; now I’m treating graphs very combinatorially, but we can bring the topology back in. From a graph $X$ we can build a topological space $|X|$ called its geometric realization. We do this by taking one point for each vertex and gluing on one copy of $[0,1]$ for each edge $e : v \to w,$ gluing the point $0$ to $v$ and the point $1$ to $w,$ and then identifying the interval for each edge $e$ with the interval for its inverse by means of the map $t \mapsto 1 - t.$

Any map of graphs gives rise to a continuous map between their geometric realizations, and we say a map of graphs is a cover if this continuous map is a covering map. For simplicity we denote the fundamental group of $|X|$ by $\pi_1(X),$ and similarly for other topological invariants of $|X|.$ However, sometimes I’ll need to distinguish between a graph $X$ and its geometric realization $|X|.$

Any connected graph $X$ has a universal cover, meaning a connected cover

$p : \widetilde{X} \to X$

that covers every other connected cover. The geometric realization of $\widetilde{X}$ is connected and simply connected. The fundamental group $\pi_1(X)$ acts as deck transformations of $\widetilde{X},$ meaning invertible maps $g : \widetilde{X} \to \widetilde{X}$ such that $p \circ g = p.$ We can take the quotient of $\widetilde{X}$ by the action of any subgroup $G \subseteq \pi_1(X)$ and get a cover $q : \widetilde{X}/G \to X.$

In particular, if we take $G$ to be the commutator subgroup of $\pi_1(X),$ we call the graph $\widetilde{X}/G$ the maximal abelian cover of the graph $X,$ and denote it by $\overline{X}.$ We obtain a cover

$q : \overline{X} \to X$

whose group of deck transformations is the abelianization of $\pi_1(X).$ This is just the first homology group $H_1(X,\mathbb{Z}).$ In particular, if the space corresponding to $X$ has $n$ holes, this is the free abelian group on
$n$ generators.

I want a concrete description of the maximal abelian cover! I’ll build it starting with the universal cover, but first we need some preliminaries on paths in graphs.

Given vertices $x,y$ in $X,$ define a path from $x$ to $y$ to be a word of edges $\gamma = e_1 \cdots e_\ell$ with $e_i : v_{i-1} \to v_i$ for some vertices $v_0, \dots, v_\ell$ with $v_0 = x$ and $v_\ell = y.$ We allow the word to be empty if and only if $x = y$; this gives the trivial path from $x$ to itself.

Given a path $\gamma$ from $x$ to $y$ we write $\gamma : x \to y,$ and we write the trivial path from $x$ to itself as $1_x : x \to x.$ We define the composite of paths $\gamma : x \to y$ and $\delta : y \to z$ via concatenation of words, obtaining a path we call $\gamma \delta : x \to z.$ We call a path from a vertex $x$ to itself a loop based at $x.$

We say two paths from $x$ to $y$ are homotopic if one can be obtained from the other by repeatedly introducing or deleting subwords of the form $e_i e_{i+1}$ where $e_{i+1} = e_i^{-1}.$ If $[\gamma]$ is a homotopy class of paths from $x$ to $y,$ we write $[\gamma] : x \to y.$ We can compose homotopy classes $[\gamma] : x \to y$ and $[\delta] : y \to z$ by setting $[\gamma] [\delta] = [\gamma \delta].$

If $X$ is a connected graph, we can describe the universal cover $\widetilde{X}$ as follows. Fix a vertex $x_0$ of $X,$ which we call the basepoint. The vertices of $\widetilde{X}$ are defined to be the homotopy classes of paths $[\gamma] : x_0 \to x$ where $x$ is arbitrary. The edges in $\widetilde{X}$ from the vertex $[\gamma] : x_0 \to x$ to the vertex $[\delta] : x_0 \to y$ are defined to be the edges $e \in E$ with $[\gamma e] = [\delta].$ In fact, there is always at most one such edge. There is an obvious map of graphs

$p : \widetilde{X} \to X$

sending each vertex $[\gamma] : x_0 \to x$ of $\widetilde{X}$ to the vertex
$x$ of $X.$ This map is a cover.

Now we are ready to construct the maximal abelian cover $\overline{X}.$ For this, we impose a further equivalence relation on paths, which is designed to make composition commutative whenever possible. However, we need to be careful. If $\gamma : x \to y$ and $\delta : x' \to y' ,$ the composites $\gamma \delta$ and $\delta \gamma$ are both well-defined if and only if $x' = y$ and $y' = x.$ In this case, $\gamma \delta$ and $\delta \gamma$ share the same starting point and share the same ending point if and only if $x = x'$ and $y = y'.$ If all four of these equations hold, both $\gamma$ and $\delta$ are loops based at $x.$ So, we shall impose the relation $\gamma \delta = \delta \gamma$ only in this case.

We say two paths are homologous if one can be obtained from another by:

• repeatedly introducing or deleting subwords $e_i e_{i+1}$ where
$e_{i+1} = e_i^{-1},$ and/or

• repeatedly replacing subwords of the form

$e_i \cdots e_j e_{j+1} \cdots e_k$

by those of the form

$e_{j+1} \cdots e_k e_i \cdots e_j$

where $e_i \cdots e_j$ and $e_{j+1} \cdots e_k$ are loops based at the same vertex.

My use of the term ‘homologous’ is a bit nonstandard here!

We denote the homology class of a path $\gamma$ by $[[ \gamma ]].$ Note that if two paths $\gamma : x \to y,$ $\delta : x' \to y'$ are homologous then $x = x'$ and $y = y'.$ Thus, the starting and ending points of a homology class of paths are well-defined, and given any path $\gamma : x \to y$ we write $[[ \gamma ]] : x \to y .$ The composite of homology classes is also well-defined if we set

$[[ \gamma ]] [[ \delta ]] = [[ \gamma \delta ]]$

We construct the maximal abelian cover of a connected graph $X$ just as we constructed its universal cover, but using homology classes rather than homotopy classes of paths. And now I’ll introduce some jargon that should make you start thinking about crystals!

Fix a basepoint $x_0$ for $X.$ The vertices of $\overline{X},$ or atoms, are defined to be the homology classes of paths $[[\gamma]] : x_0 \to x$ where $x$ is arbitrary. Any edge of $\overline{X},$ or bond, goes from some atom $[[ \gamma]] : x_0 \to x$ to the some atom $[[ \delta ]] : x_0 \to y.$ The bonds from $[[ \gamma]]$ to $[[ \delta ]]$ are defined to be the edges $e \in E$ with $[[ \gamma e ]] = [[ \delta ]].$ There is at most one bond between any two atoms. Again we have a covering map

$q : \overline{X} \to X .$

The homotopy classes of loops based at $x_0$ form a group, with composition as the group operation. This is the fundamental group $\pi_1(X)$ of the graph $X.$ This is isomorphic as the fundamental group of the space associated to $X.$ By our construction of the universal cover, $\pi_1(X)$ is also the set of vertices of $\widetilde{X}$ that are mapped to $x_0$ by $p.$ Furthermore, any element $[\gamma] \in \pi_1(X)$ defines a deck transformation of $\widetilde{X}$ that sends each vertex $[\delta] : x_0 \to x$ to the vertex $[\gamma] [\delta] : x_0 \to x.$

Similarly, the homology classes of loops based at $x_0$ form a group with composition as the group operation. Since the additional relation used to define homology classes is precisely that needed to make composition of homology classes of loops commutative, this group is the abelianization of $\pi_1(X).$ It is therefore isomorphic to the first homology group $H_1(X,\mathbb{Z})$ of the geometric realization of $X.$

By our construction of the maximal abelian cover, $H_1(X,\mathbb{Z})$ is also the set of vertices of $\overline{X}$ that are mapped to $x_0$ by $q.$ Furthermore, any element $[[\gamma]] \in H_1(X,\mathbb{Z})$ defines a deck transformation of $\overline{X}$ that sends each vertex $[[\delta]] : x_0 \to x$ to the vertex $[[\gamma]] [[\delta]] : x_0 \to x.$

So, it all works out! The fundamental group $\pi_1(X)$ acts as deck transformations of the universal cover, while the first homology group $H_1(X,\mathbb{Z})$ acts as deck transformations of the maximal abelian cover.

Puzzle for experts: what does this remind you of in Galois theory?

We’ll get back to crystals next time.

Part 1 – the basic idea.

Part 2 – the maximal abelian cover of a graph.

Part 3 – constructing topological crystals.

Part 4 – examples of topological crystals.

## Topological Crystals (Part 1)

22 July, 2016

A while back, we started talking about crystals:

• John Baez, Diamonds and triamonds, Azimuth, 11 April 2016.

In the comments on that post, a bunch of us worked on some puzzles connected to ‘topological crystallography’—a subject that blends graph theory, topology and mathematical crystallography. You can learn more about that subject here:

• Tosio Sunada, Crystals that nature might miss creating, Notices of the AMS 55 (2008), 208–215.

I got so interested that I wrote this paper about it, with massive help from Greg Egan:

• John Baez, Topological crystals.

I’ll explain the basic ideas in a series of posts here.

First, a few personal words.

I feel a bit guilty putting so much work into this paper when I should be developing network theory to the point where it does our planet some good. I seem to need a certain amount of beautiful pure math to stay sane. But this project did at least teach me a lot about the topology of graphs.

For those not in the know, applying homology theory to graphs might sound fancy and interesting. For people who have studied a reasonable amount of topology, it probably sounds easy and boring. The first homology of a graph of genus $g$ is a free abelian group on $g$ generators: it’s a complete invariant of connected graphs up to homotopy equivalence. Case closed!

But there’s actually more to it, because studying graphs up to homotopy equivalence kills most of the fun. When we’re studying networks in real life we need a more refined outlook on graphs. So some aspects of this project might pay off, someday, in ways that have nothing to do with crystallography. But right now I’ll just talk about it as a fun self-contained set of puzzles.

I’ll start by quickly sketching how to construct topological crystals, and illustrate it with the example of graphene, a 2-dimensional form of carbon:

I’ll precisely state our biggest result, which says when this construction gives a crystal where the atoms don’t bump into each other and the bonds between atoms don’t cross each other. Later I may come back and add detail, but for now you can find details in our paper.

### Constructing topological crystals

The ‘maximal abelian cover’ of a graph plays a key role in Sunada’s work on topological crystallography. Just as the universal cover of a connected graph $X$ has the fundamental group $\pi_1(X)$ as its group of deck transformations, the maximal abelian cover, denoted $\overline{X},$ has the abelianization of $\pi_1(X)$ as its group of deck transformations. It thus covers every other connected cover of $X$ whose group of deck transformations is abelian. Since the abelianization of $\pi_1(X)$ is the first homology group $H_1(X,\mathbb{Z}),$ there is a close connection between the maximal abelian cover and homology theory.

In our paper, Greg and I prove that for a large class of graphs, the maximal abelian cover can naturally be embedded in the vector space $H_1(X,\mathbb{R}).$ We call this embedded copy of $\overline{X}$ a ‘topological crystal’. The symmetries of the original graph can be lifted to symmetries of its topological crystal, but the topological crystal also has an $n$-dimensional lattice of translational symmetries. In 2- and 3-dimensional examples, the topological crystal can serve as the blueprint for an actual crystal, with atoms at the vertices and bonds along the edges.

The general construction of topological crystals was developed by Kotani and Sunada, and later by Eon. Sunada uses ‘topological crystal’ for an even more general concept, but we only need a special case.

Here’s how it works. We start with a graph $X.$ This has a space $C_0(X,\mathbb{R})$ of 0-chains, which are formal linear combinations of vertices, and a space $C_1(X,\mathbb{R})$ of 1-chains, which are formal linear combinations of edges. There is a boundary operator

$\partial \colon C_1(X,\mathbb{R}) \to C_0(X,\mathbb{R})$

This is the linear operator sending any edge to the difference of its two endpoints. The kernel of this operator is called the space of 1-cycles, $Z_1(X,\mathbb{R}).$ There is an inner product on the space of 1-chains such that edges form an orthonormal basis. This determines an orthogonal projection

$\pi \colon C_1(X,\mathbb{R}) \to Z_1(X,\mathbb{R})$

For a graph, $Z_1(X,\mathbb{R})$ is isomorphic to the first homology group $H_1(X,\mathbb{R}).$ So, to obtain the topological crystal of $X,$ we need only embed its maximal abelian cover $\overline{X}$ in $Z_1(X,\mathbb{R}).$ We do this by embedding $\overline{X}$ in $C_1(X,\mathbb{R})$ and then projecting it down via $\pi.$

To accomplish this, we need to fix a basepoint for $X.$ Each path $\gamma$ in $X$ starting at this basepoint determines a 1-chain $c_\gamma.$ These 1-chains correspond to the vertices of $\overline{X}.$ The graph $\overline{X}$ has an edge from $c_\gamma$ to $c_{\gamma'}$ whenever the path $\gamma'$ is obtained by adding an extra edge to $\gamma.$ This edge is a straight line segment from the point $c_\gamma$ to the point $c_{\gamma'}.$

The hard part is checking that the projection $\pi$ maps this copy of $\overline{X}$ into $Z_1(X,\mathbb{R})$ in a one-to-one manner. In Theorems 6 and 7 of our paper we prove that this happens precisely when the graph $X$ has no ‘bridges’: that is, edges whose removal would disconnect $X.$

Kotani and Sunada noted that this condition is necessary. That’s actually pretty easy to see. The challenge was to show that it’s sufficient! For this, our main technical tool is Lemma 5, which for any path $\gamma$ decomposes the 1-chain $c_\gamma$ into manageable pieces.

We call the resulting copy of $\overline{X}$ embedded in $Z_1(X,\mathbb{R})$ a topological crystal.

Let’s see how it works in an example!

Take $X$ to be this graph:

Since $X$ has 3 edges, the space of 1-chains is 3-dimensional. Since $X$ has 2 holes, the space of 1-cycles is a 2-dimensional plane in this 3-dimensional space. If we consider paths $\gamma$ in $X$ starting at the red vertex, form the 1-chains $c_\gamma,$ and project them down to this plane, we obtain the following picture:

Here the 1-chains $c_\gamma$ are the white and red dots. These are the vertices of $\overline{X},$ while the line segments between them are the edges of $\overline{X}.$ Projecting these vertices and edges onto the plane of 1-cycles, we obtain the topological crystal for $X.$ The blue dots come from projecting the white dots onto the plane of 1-cycles, while the red dots already lie on this plane. The resulting topological crystal provides the pattern for graphene:

That’s all there is to the basic idea! But there’s a lot more to say about it, and a lot of fun examples to look at: diamonds, triamonds, hyperquartz and more.