A twist that carries one loop to another

The loops on a torus that never cross themselves are the classes $(p, q)$ with no common factor — slopes in lowest terms — together with the loops that shrink to a point. There are infinitely many of them, and on the square they look quite different: $(1, 0)$ is one horizontal line, $(5, 2)$ is seven parallel slanted pieces.

They are nevertheless all the same loop. There is a symmetry of the torus — a way of moving every point so that nothing is torn and nothing is glued that was not glued before — carrying any one of them to any other. The symmetries in question are not rotations or reflections of a rigid doughnut. They are built from one move, studied by Max Dehn in the 1930s: cut the torus along a loop, turn one side of the cut once round, and glue it back.

This essay draws that move, follows what it does to the two counts of every loop, and finds that two twists generate every symmetry of the torus there is — and that the group they generate has appeared before, as the pair of matrices that grows the tree of fractions.

The twist, drawn #

Take the loop of class $(1, 0)$, the horizontal line, and thicken it into a band running once round the torus. Cut the torus along the bottom edge of the band. The two sides of the cut are both copies of the same circle, and turning one of them a full revolution before gluing it back to the other brings every point back to a point it was glued to before — so the result is the same torus, with every point moved.

Inside the band the move is a shear. A point at the bottom of the band does not move, a point at the top moves once round, and a point in between moves the corresponding fraction of the way; outside the band nothing moves at all. The vertical loop of class $(0, 1)$ enters the band at the bottom, is dragged once round the horizontal direction as it climbs through, and leaves at the top one full turn along. It now crosses the vertical edge once and the horizontal edge once: it is a loop of class $(1, 1)$.

Nothing was torn and nothing was glued that had not been glued before, so a loop that did not cross itself still does not. The twisted loop is bent where it passes through the band, but it is a simple loop, and so it is deformable to the straight loop of its class. The twist has carried one simple loop to another.

What the twist does to every class #

A loop of class $(p, q)$ passes up through the band $q$ times, counted with sign, and each passage drags it once round. So it picks up $q$ extra crossings of the vertical edge and no extra crossings of the horizontal one:

$$(p, q) \;\longmapsto\; (p + q,\ q), \qquad \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} p \\ q \end{pmatrix}.$$

Twisting again adds $q$ again: $(1, 1)$ goes to $(2, 1)$ and then $(3, 1)$, and in general the $k$-th twist of $(p, q)$ is $(p + kq, q)$. Starting from the vertical loop $(0, 1)$, the twists produce $(1, 1), (2, 1), (3, 1), \dots$ — infinitely many different simple loops from one, none of which can be deformed into any other. The horizontal loop itself, with $q = 0$, is fixed: it lies along the band, never passes through it, and the twist moves it only along itself.

The twist along the vertical loop does the same with the roles exchanged, turning in the same sense: a loop crossing the vertical band $p$ times has $p$ subtracted from its second count, $(p, q) \mapsto (p, q - p)$, which is the matrix with rows $(1, 0)$ and $(-1, 1)$. The minus sign is the orientation: seen from the vertical loop, a twist that turns the same way as the horizontal one drags crossings downward.

Euclid’s algorithm, done with twists #

The twists can also be run backwards. The inverse of the horizontal twist subtracts: $(p, q) \mapsto (p - q, q)$. The vertical twist subtracts the other way. And a pair of whole numbers reduced by repeatedly subtracting the smaller from the larger is Euclid’s algorithm.

The figure walks the class $(5, 3)$ down: $(5, 3) \to (2, 3) \to (2, 1) \to (1, 1) \to (1, 0)$, four twists, each subtracting the smaller count from the larger. Euclid’s algorithm on two numbers with no common factor always ends at 1 and 0, so every coprime class walks down to $(1, 0)$, and every class on the way is coprime too, since subtracting one count from the other changes no common factor.

Run backwards, the same four twists carry the horizontal loop to a simple loop of class $(5, 3)$. So every simple loop on the torus that does not shrink away is the image of the horizontal loop under some symmetry — a product of twists. Up to symmetry, the torus has exactly one such loop, and the whole arithmetic of the previous classification measures where that one loop has been moved relative to a fixed pair of edges.

The same subtraction reads a node’s address in the tree of fractions, and this is not a coincidence of method. The horizontal twist is the matrix that tree calls $R$, and the vertical twist run backwards is its $L$. The tree was an orbit of a pair of fractions under two matrices; here the same two matrices are two symmetries of a surface, and the tree’s fractions are the torus’s simple loops. Euclid’s game, where two players take turns subtracting a multiple of the smaller number from the larger, is the same walk again, with the choice of how many steps to take at once left to the player.

Every symmetry of the torus #

Every symmetry of the torus acts on classes, sending $(p, q)$ to some $(p', q')$, and the action is linear, since a symmetry that carries the loop of class $x$ to the loop of class $x'$ carries a loop going round $x$ twice to one going round $x'$ twice. So each symmetry gives a two-by-two matrix of whole numbers.

That matrix has determinant one, and the reason is the other half of the classification. Two loops of classes $(p, q)$ and $(r, s)$ must meet $|ps - qr|$ times, counted with sign, and a symmetry that does not turn the surface over preserves those signed crossings. The determinant of the matrix is exactly the factor by which it multiplies $ps - qr$, so it must be $+1$. The matrices of the twists have determinant one, as they must.

The converse is true as well, and the reason is almost visible. A matrix of whole numbers with determinant one is a linear map of the plane that carries the lattice of whole-number points onto itself, so it passes to the square with its edges glued, and gives a symmetry of the torus. And two symmetries acting by the same matrix can be deformed into one another — that half is a theorem, quoted here rather than proved. Together: the symmetries of the torus that do not turn it over, counted up to deformation, are exactly the whole-number matrices of determinant one, the group written $\mathrm{SL}(2, \mathbb{Z})$ and called the modular group. Topologists call it the torus’s mapping class group.

Because Euclid’s walk reaches $(1, 0)$ from any coprime class, and because a matrix of determinant one is determined by where it sends $(1, 0)$ up to a twist along the image, the two twists generate the whole group. William Lickorish proved in the early 1960s that twists generate the symmetries of every closed orientable surface; for the torus, two suffice.

A symmetry is a new choice of edges #

There is a way to see the whole of the last section in one picture, without any twisting. The square the torus was built from had two edges, and those edges are two simple loops meeting exactly once — the horizontal and vertical loops. Any other pair of simple loops meeting exactly once would have done just as well: cut the torus along both and it opens out into a square again, with those two loops as its edges.

So a symmetry of the torus is a change of which pair of loops is called the edges. It sends the old pair to a new pair meeting once, and the new pair’s classes, written as the columns of a matrix, have determinant $\pm 1$ — plus one if the symmetry keeps the surface’s sides, since that is exactly the condition that the new loops meet once with the same sign as the old. Conversely any such pair of columns names a pair of loops meeting once, and cutting along them builds the symmetry. It is the same map written in a different basis, with the extra demand that the new basis consist of whole numbers and span the same lattice.

That is why the determinant appeared and why nothing else could have. A matrix of whole numbers with determinant 2 would send the square to a parallelogram of twice the area, covering the torus twice over — a cover rather than a symmetry, and a map of the torus onto itself that is two-to-one.

Two twists, and symmetries of finite order #

Each twist has infinite order: twisting again and again produces infinitely many different loops from $(0, 1)$, as the second figure showed. But products of the two can have finite order, and the smallest ones are the rigid symmetries of special tori.

Twist horizontally, then vertically, then horizontally again. The product is the matrix with rows $(0, 1)$ and $(-1, 0)$ — a quarter-turn of the lattice, which exchanges the horizontal and vertical loops. On a square torus, one built from a square rather than a parallelogram, it is a genuine rotation. Applied four times it is the identity: the loop of class $(2, 1)$ is carried round through $(1, -2)$, $(-2, -1)$ and $(-1, 2)$ and back. Applied twice it sends every class to its negative — every loop to itself run backwards — which on a doughnut is the half-turn about an axis through the hole.

Doing the twists in the other order gives the same symmetry: horizontal-vertical-horizontal equals vertical-horizontal-vertical. That is the braid relation, and it holds for twists along any two loops that meet exactly once — on any surface, not only this one.

Twist vertically and then horizontally, just once each, and the product has order six: the horizontal loop is carried through $(0, -1)$, $(-1, -1)$, $(-1, 0)$, $(0, 1)$ and $(1, 1)$ and back. On a torus built from a rhombus of two equilateral triangles — the hexagonal torus — this is a rotation by a sixth of a turn. The two twists, neither of which ever repeats, compose to a symmetry that repeats every sixth time, and the whole group is built from these two finite pieces: the modular group is generated by an element of order four and one of order six whose squares and cubes agree.

Three kinds of symmetry, told apart by a trace #

The symmetries of the torus fall into three kinds, and the sum of the diagonal entries of the matrix — the trace — says which.

Trace between −2 and 2, excluding the ends: the symmetry has finite order — three, four or six — and it is a rigid rotation of some flat torus. The quarter-turn has trace 0 and the sixth-turn trace 1.

Trace exactly 2 or −2: the symmetry fixes a simple loop, up to reversal, and is a power of the twist along that loop, possibly followed by the half-turn. The horizontal twist has trace 2 and fixes the horizontal loop.

Trace greater than 2 in size: the symmetry fixes no loop at all. Its matrix has two real eigenvalues, one larger than 1 and one smaller, and the symmetry stretches the torus in one irrational direction and shrinks it in another. The simplest is the product of a horizontal twist and a vertical twist run backwards, the matrix with rows $(2, 1)$ and $(1, 1)$, which stretches by the square of the golden ratio and shrinks by its inverse. Every loop, repeatedly transformed, grows in length by that factor each time in the long run. This is the first appearance of the pattern William Thurston established for every surface: every symmetry is, up to deformation, of finite order, or reducible along a loop, or of this stretching kind.

What the pictures cannot show #

Only one twist is drawn as a moving picture. The horizontal twist is drawn as a shear of the square and its effect on each loop is counted from the drawing; the vertical twist is used through its matrix, and the finite-order symmetries are drawn by their action on straight loops, not as deformations of the square.

The twist is drawn with a band of one fixed width. Any width gives the same symmetry up to deformation, and a narrower band would make the bend sharper without changing the class of any loop; the choice is for legibility only.

The step from matrices back to symmetries is quoted. That two symmetries acting by the same matrix can be deformed into one another is a theorem about the torus, and nothing here proves it. Without it the matrices would be a shadow of the symmetries rather than the whole of them.

And the stretching symmetries are described, not drawn. Their matrices are listed and their eigenvalues computed in the prose; no figure shows a loop being stretched, and the claim that every loop grows by the same factor in the long run is stated from the eigenvalues rather than measured.

Beyond the torus #

Two questions about the higher homotopy groups remain open here: how the loops of a space act on them, and the Hopf map. The twist points in a third direction — from classifying loops to classifying the symmetries that move them. On surfaces of higher genus the same construction gives the mapping class group, generated by twists along finitely many loops, and far richer than $\mathrm{SL}(2, \mathbb{Z})$; the relations among its generators, and the braid relation in particular, lead to the braid groups and to the study of knots through the symmetries of their complements.

On a surface with more handles the simple loops are no longer all alike: a loop that cuts the surface into two pieces cannot be carried to one that leaves it whole, since cutting is preserved by any symmetry. Every closed surface is a sphere with handles, and the classification of its simple loops up to symmetry is a list of the ways of cutting it — finite, and read off from the genus of the pieces. The torus is the one surface where the list has a single entry.

One loop, moved everywhere #

Cutting a torus along a loop and regluing it with a full turn is a symmetry, and on classes it adds one count to the other. Run backwards, twists perform Euclid’s algorithm on the two counts, so every loop that never crosses itself is the image of one loop under twists, and the symmetries the twists generate are exactly the whole-number matrices of determinant one. Two twists of infinite order compose to rotations of order four and six, and the trace sorts every symmetry into rotating, twisting or stretching.

The classification of simple loops found infinitely many of them and a rule for which classes hold one. This one finds that they are one loop in infinitely many positions, and that the rule was a description of the positions. The arithmetic did not go away; it moved from the loops to the symmetries, where it became a group of matrices that had already appeared growing a tree of fractions.

When a space has many objects of one kind, look for the symmetries that carry them to one another — the classification that remains is of positions, and it is usually arithmetic.