Every surface is sewn from pants

The classification in every surface is a sphere with handles builds a surface up: start from a sphere and attach handles. This essay runs the other way, and cuts a surface down. The question is what the smallest pieces are, and the answer is a shape with a comic name and a precise definition: a pair of pants, a sphere with three discs removed — a waist and two legs.

The figure below is every way to cut the surface with two handles into pants. There are two. On the left, three circles cut it into two pieces, and one of those circles — the middle one — separates the surface into two halves. On the right, three circles again, none of which separates anything on its own. Each coloured region is one pair of pants. Every cutting of the two-handled surface into pants looks like one of these two, after the surface is suitably deformed, and each uses exactly three circles and two pants.

That the numbers agree between the two is not a coincidence, and nor is the count of two. The first is forced by one number the surface carries; the second comes from listing graphs.

Why pants are the atoms #

Cut a surface along a circle and the pieces are simpler, and it is worth asking where that stops. A disc cannot be cut into anything smaller in an interesting way, and nor can an annulus: every circle drawn in either one either bounds a disc or runs parallel to the boundary. Those two pieces have Euler characteristic $1$ and $0$.

A pair of pants has characteristic $-1$ — the sphere’s $2$, less one for each of the three removed discs. It is the simplest surface with negative characteristic, and it has the same property as the disc and the annulus: every circle in it bounds a disc or runs parallel to one of its three cuffs. There is nowhere left to cut. A pair of pants is the smallest surface on which the characteristic is negative, and that is why it is the atom for every surface whose characteristic is.

Any other surface with negative characteristic does contain a circle that is neither of those kinds, and cutting along it leaves pieces that still have negative characteristic, because cutting along a circle does not change the total. Keep cutting and the pieces must eventually be pants: each cut reduces a count of how much room there is left for circles, which cannot fall forever.

The number of pieces is not a choice #

The Euler characteristic is additive under cutting along circles. A circle has characteristic zero — as many vertices as edges — so cutting along one and counting the two new boundary circles leaves the total unchanged. The characteristics of the pieces add up to the characteristic of the surface.

The closed surface with $g$ handles has characteristic $2 - 2g$, the count that the classification runs on. Every pair of pants contributes $-1$. So there are $2g - 2$ pants in any decomposition at all: two for the two-handled surface, four for three handles, eighteen for ten.

The circles follow by counting cuffs. Each pair of pants has three, so there are $3(2g - 2)$ cuffs in all, and each cutting circle accounts for two of them — one on each side. So there are $3g - 3$ circles.

A surface with boundary, the case the third number a surface needs added, obeys the same arithmetic with one change. Its characteristic is $2 - 2g - b$, so it takes $2g - 2 + b$ pants. Its own boundary circles use up one cuff each without being cut, so the cutting circles number $3g - 3 + b$. The figure checks the cuff count in every cell — three per pair of pants must equal two per cutting circle plus one per boundary circle — and leaves blank the four surfaces with no negative characteristic: the sphere, the disc, the annulus and the torus, which are made of discs and annuli instead.

The number $3g - 3 + b$ is the most disjoint circles a surface can hold without two of them running parallel or one of them bounding a disc. Any collection of that many is automatically a decomposition into pants, and no collection can be larger. It is a statement about room, and the characteristic measures the room.

A decomposition is a graph #

The counting says nothing about how the pants fit together, and the first figure shows that they can fit in different ways. The way to see all the possibilities is to draw each decomposition as a graph.

Put a vertex for each pair of pants and an edge for each cutting circle, joining the two pants it separates. Every vertex has three ends, one per cuff. An edge can join a vertex to itself, when both sides of a circle are cuffs of the same pair of pants, and two vertices can be joined by more than one edge. The graph has $2g - 2$ vertices and $3g - 3$ edges, so it has $g$ independent cycles — one for each handle.

The pictures in these figures are the reverse construction. Thicken the graph into a tube and its skin is the surface: every vertex becomes a junction of three tubes, which is a pair of pants, and every edge becomes a tube, cut round its middle by the circle the edge stands for. Any graph with three ends at every vertex and $g$ independent cycles thickens this way into the surface with $g$ handles, cut into pants.

Two decompositions are the same up to a deformation of the surface exactly when their graphs are the same up to relabelling the vertices. So counting the types of decomposition is counting these graphs, and a computer can do it without insight: fill in how many edges join each pair of vertices, keeping every vertex at three ends, discard the disconnected results, and identify two graphs whenever some relabelling carries one to the other. For three handles that search finds five, drawn above.

Two, five, seventeen #

For two handles the search can be done by hand, and doing it shows why the answer is two. There are two vertices and three edges. If either vertex carries a loop, that loop uses two of its three ends and the third must go to the other vertex, which then has two ends left and nowhere to send them but a loop of its own: the dumbbell. If neither carries a loop, all three edges run between the two vertices: the theta. There is no third possibility.

The counts grow quickly — two, five, seventeen — and the structure inside them is readable from the graphs. A cutting circle separates the surface exactly when its edge is a bridge of the graph, an edge whose removal disconnects it: cutting the tube round a bridge leaves the thickened graph in two pieces, and cutting round any other edge does not.

At one extreme is a decomposition with no separating circle at all, which is a graph with no bridge. For two handles it is the theta on the right of the head figure, for three handles there are two, and for four there are five. At the other extreme, the graph is a tree with a loop at every leaf, and every edge of the tree is a bridge. There are $g$ leaves, each carrying one handle as a loop, and $2g - 3$ tree edges, so at most $2g - 3$ circles can separate at once — one for two handles, three for three, five for four. The figure checks that bound on every type it lists.

These are counts up to the symmetries of the surface. Counted without that identification, a surface has infinitely many decompositions: any one can be twisted by a homeomorphism of the surface into another, and the homeomorphisms form an infinite group. The types are the finitely many orbits, and how many there are for general genus is a question the third number a surface needs left open. None is known: the numbers of these graphs come from enumeration, and they grow faster than any exponential in $g$, much as the number of ways to pair a polygon’s edges does.

One circle at a time #

A decomposition can be changed into another by removing one cutting circle and putting a different one in its place. The circle removed was the boundary between two pieces, and removing it merges them into one surface of characteristic $-2$ — two pants glued along a cuff. That surface is either a sphere with four holes or, if the two pieces were the same pair of pants glued to itself, a torus with one hole.

Either way, the merged piece has room for exactly one circle, and many choices of it. Put in a different one and the surface is again cut into pants. That is an elementary move.

On the graph, a move on a four-holed sphere is a Whitehead move: collapse the edge between the two pants, leaving a vertex with four ends, and pull it apart again the other way. There are two other ways to split four ends into two pairs, and each gives a graph. On the theta, collapse any one of the three parallel edges: of the two new splittings, one gives the theta back with its edges relabelled, and the other turns the remaining two parallel edges into loops, one at each end — the dumbbell. The figure checks that the move goes both ways, from theta to dumbbell and back.

Allen Hatcher and William Thurston proved in 1980 that any two decompositions of a surface are joined by a finite chain of elementary moves. They were after something else — a finite description of the surface’s group of symmetries — and the connectedness of the moves was the tool that produced it.

The five genus-3 types show the theorem’s shadow on types. From the decomposition with three separating circles, moves lead one step at a time to the one with none, passing through each type in between. Each link was found by applying every Whitehead move to every type and identifying the result, and the figure checks that the links connect everything. Hatcher and Thurston’s theorem is the much stronger statement about the decompositions themselves, not their types, where there are infinitely many and the graph of moves between them — the pants graph — is an infinite, connected object with a geometry of its own.

One-sided surfaces break the circle count #

Everything so far was about two-sided surfaces, and the one-sided half of the classification is instructive because one of the two counts survives and the other does not.

A one-sided surface contains circles of a kind the two-sided ones lack: circles whose neighbourhood is a Möbius band rather than an annulus. Cut along such a circle and only one new boundary circle appears, not two, because the Möbius band’s edge runs round twice. The smallest one-sided closed surface, a disc sewn to a Möbius band, is exactly that: cut along the band’s core and a disc is left, with one boundary circle.

The pants count is unaffected, since a Möbius band has characteristic zero just as an annulus does: the surface with $k$ cross-caps has characteristic $2 - k$ and falls into $k - 2$ pairs of pants. But the cuffs can now be paid for in two currencies. A two-sided cutting circle uses two cuffs and a one-sided circle uses one, so the number of circles is not fixed.

The surface with three cross-caps shows both extremes at once. It is one pair of pants with three cuffs. Cap every cuff with a Möbius band and the surface is cut by three one-sided circles. Or glue two of the cuffs to each other with a flip, which makes a Klein bottle with one hole, and cap the third with a Möbius band: the same surface, now cut by one two-sided circle and one one-sided circle — two circles instead of three. On one-sided surfaces the characteristic fixes the pieces and not the cuts, and the dimension count that made the two-sided pants so useful has to be redone for each pattern.

What the pieces are for #

The reason pants decompositions organise so much of two-dimensional geometry is that each pair of pants is rigid in a useful way. Give a surface a geometry of constant negative curvature, as every surface with two or more handles can be given, and the cutting circles can be pulled tight into shortest curves. A pair of pants in that geometry is then determined completely by the three lengths of its cuffs: choose any three positive numbers and there is exactly one such pair of pants, and every one has the same area, $2\pi$, by the Gauss–Bonnet theorem that ties curvature to the characteristic.

So a geometry on the whole surface is described by $3g - 3$ lengths — one per cutting circle — and $3g - 3$ angles of twist, saying how far each pair of cuffs is rotated before they are sewn together. That is $6g - 6$ numbers, and they are the Fenchel–Nielsen coordinates: a complete, independent description of every possible shape of the surface. The dimension of the space of shapes, $6g - 6$, is read straight off the pants count. For the two-handled surface it is six; for the torus, which has no pants decomposition, the space of shapes is two-dimensional and has to be described another way.

Lipman Bers proved that every surface of genus $g$ with such a geometry has a decomposition whose circles are all shorter than a bound depending only on $g$. So among the infinitely many decompositions of any particular shape of surface, some short one always exists, and it is the natural coordinate chart for that shape. The pants graph, with moves as its steps, was shown by Jeffrey Brock in 2003 to have the same large-scale geometry as the space of shapes itself, under one of its natural metrics.

What the figures cannot show #

The thickened graphs are not the surface’s own shape. They show which pants are sewn to which along which circle, and nothing about lengths or twists. The same drawing stands for every geometry with that pattern, and the Fenchel–Nielsen coordinates are what distinguish them.

The census stops at four handles. The search relabels every candidate graph in every way, which is $720$ relabellings for six vertices and forty thousand for eight. The counts for five and six handles are known from better methods, and the method here is chosen because it cannot miss a type or count one twice, not because it scales.

And the types hide the twisting. Two decompositions of the same type differ by a homeomorphism of the surface, which can wind a circle round the surface many times. The figures draw one representative of each type and say nothing about how many decompositions share it, which is infinitely many.

Still open: counting decompositions by length #

The types are counted here by exhaustion, and no formula gives them for general genus. A sharper question counts actual decompositions rather than types, on a surface with a fixed geometry: how many decompositions of a given type have cutting circles of total length at most $L$? Maryam Mirzakhani showed in 2008 that the number grows like a constant times $L^{6g-6}$ — the dimension of the space of shapes again — and computed the constants by integrating over that space, but how the constants depend on the particular shape, and how fast the count settles to its growth rate, are still being worked out.

The same arithmetic that fixes the pants count asks the next question about pieces: what is the fewest number of corners a surface can be built from out of triangles, where the characteristic sets only a floor and search finds three surfaces that cannot meet it. And the characteristic’s additivity is what makes a covering multiply the count: a surface covering a genus-$g$ surface $d$ times has $d(2g - 2)$ pants’ worth of characteristic, whatever its decomposition looks like.

What is worth carrying away #

A count that can be proved from an invariant should be. How many pieces and how many cuts are fixed by the Euler characteristic before any cutting is done, and the figures confirm it on every type rather than establishing it.

The habit worth keeping is to redraw a decomposition as a graph. Everything combinatorial about how the pieces fit — which circles separate, how many types there are, which moves connect them — becomes a question about vertices and edges, and a question a search can settle.