Periodic Patterns in Peptide Masses

6 April, 2017

Gheorghe Craciun is a mathematician at the University of Wisconsin who recently proved the Global Attractor Conjecture, which since 1974 was the most famous conjecture in mathematical chemistry. This week he visited U. C. Riverside and gave a talk on this subject. But he also told me about something else—something quite remarkable.

The mystery

A peptide is basically a small protein: a chain of made of fewer than 50 amino acids. If you plot the number of peptides of different masses found in various organisms, you see peculiar oscillations:

These oscillations have a frequency of about 14 daltons, where a ‘dalton’ is roughly the mass of a hydrogen atom—or more precisely, 1/12 the mass of a carbon atom.

Biologists had noticed these oscillations in databases of peptide masses. But they didn’t understand them.

Can you figure out what causes these oscillations?

It’s a math puzzle, actually.

Next I’ll give you the answer, so stop looking if you want to think about it first.

The solution

Almost all peptides are made of 20 different amino acids, which have different masses, which are almost integers. So, to a reasonably good approximation, the puzzle amounts to this: if you have 20 natural numbers m_1, ... , m_{20}, how many ways can you write any natural number N as a finite ordered sum of these numbers? Call it F(N) and graph it. It oscillates! Why?

(We count ordered sums because the amino acids are stuck together in a linear way to form a protein.)

There’s a well-known way to write down a formula for F(N). It obeys a linear recurrence:

F(N) = F(N - m_1) + \cdots + F(N - m_{20})

and we can solve this using the ansatz

F(N) = x^N

Then the recurrence relation will hold if

x^N = x^{N - m_1} + x^{N - m_2} + \dots + x^{N - m_{20}}

for all N. But this is fairly easy to achieve! If m_{20} is the biggest mass, we just need this polynomial equation to hold:

x^{m_{20}} = x^{m_{20} - m_1} + x^{m_{20} - m_2} + \dots + 1

There will be a bunch of solutions, about m_{20} of them. (If there are repeated roots things get a bit more subtle, but let’s not worry about.) To get the actual formula for F(N) we need to find the right linear combination of functions x^N where x ranges over all the roots. That takes some work. Craciun and his collaborator Shane Hubler did that work.

But we can get a pretty good understanding with a lot less work. In particular, the root x with the largest magnitude will make x^N grow the fastest.

If you haven’t thought about this sort of recurrence relation it’s good to look at the simplest case, where we just have two masses m_1 = 1, m_2 = 2. Then the numbers F(N) are the Fibonacci numbers. I hope you know this: the Nth Fibonacci number is the number of ways to write N as the sum of an ordered list of 1’s and 2’s!

1

1+1,   2

1+1+1,   1+2,   2+1

1+1+1+1,   1+1+2,   1+2+1,   2+1+1,   2+2

If I drew edges between these sums in the right way, forming a ‘family tree’, you’d see the connection to Fibonacci’s original rabbit puzzle.

In this example the recurrence gives the polynomial equation

x^2 = x + 1

and the root with largest magnitude is the golden ratio:

\Phi = 1.6180339...

The other root is

1 - \Phi = -0.6180339...

With a little more work you get an explicit formula for the Fibonacci numbers in terms of the golden ratio:

\displaystyle{ F(N) = \frac{1}{\sqrt{5}} \left( \Phi^{N+1} - (1-\Phi)^{N+1} \right) }

But right now I’m more interested in the qualitative aspects! In this example both roots are real. The example from biology is different.

Puzzle 1. For which lists of natural numbers m_1 < \cdots < m_k are all the roots of

x^{m_k} = x^{m_k - m_1} + x^{m_k - m_2} + \cdots + 1

real?

I don’t know the answer. But apparently this kind of polynomial equation always one root with the largest possible magnitude, which is real and has multiplicity one. I think it turns out that F(N) is asymptotically proportional to x^N where x is this root.

But in the case that’s relevant to biology, there’s also a pair of roots with the second largest magnitude, which are not real: they’re complex conjugates of each other. And these give rise to the oscillations!

For the masses of the 20 amino acids most common in life, the roots look like this:

The aqua root at right has the largest magnitude and gives the dominant contribution to the exponential growth of F(N). The red roots have the second largest magnitude. These give the main oscillations in F(N), which have period 14.28.

For the full story, read this:

• Shane Hubler and Gheorghe Craciun, Periodic patterns in distributions of peptide masses, BioSystems 109 (2012), 179–185.

Most of the pictures here are from this paper.

My main question is this:

Puzzle 2. Suppose we take many lists of natural numbers m_1 < \cdots < m_k and draw all the roots of the equations

x^{m_k} = x^{m_k - m_1} + x^{m_k - m_2} + \cdots + 1

What pattern do we get in the complex plane?

I suspect that this picture is an approximation to the answer you’d get to Puzzle 2:

If you stare carefully at this picture, you’ll see some patterns, and I’m guessing those are hints of something very beautiful.

Earlier on this blog we looked at roots of polynomials whose coefficients are all 1 or -1:

The beauty of roots.

The pattern is very nice, and it repays deep mathematical study. Here it is, drawn by Sam Derbyshire:


But now we’re looking at polynomials where the leading coefficient is 1 and all the rest are -1 or 0. How does that change things? A lot, it seems!

By the way, the 20 amino acids we commonly see in biology have masses ranging between 57 and 186. It’s not really true that all their masses are different. Here are their masses:

57, 71, 87, 97, 99, 101, 103, 113, 113, 114, 115, 128, 128, 129, 131, 137, 147, 156, 163, 186

I pretended that none of the masses m_i are equal in Puzzle 2, and I left out the fact that only about 1/9th of the coefficients of our polynomial are nonzero. This may affect the picture you get!


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.”

The speakers include John Baez, Sophie de Buyl, Massimiliano Esposito, Arren Bar-Even, Christoff Flamm, Ronan Fleming, Christian Gaspard, Daniel Merkle, Philippe Nge, Thomas Ouldridge, Luca Peliti, Matteo Polettini, Hong Qian, Stefan Schuster, Alexander Skupin, Pieter Rein ten Wolde. 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.

networks_compositionality


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’:

Monoidal categories of networks.

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


k4_crystal

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.

Read the whole series

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


k4_crystal

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!

Read the whole series

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

k4_crystal

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.

Read the whole series

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.