Topology — page 4

Seven graphs that must link

Six points joined in every way cannot be placed in space without two disjoint triangles hooking together. Trade any triangle for a three-pointed star, or a star back for a triangle, and the property survives; doing it in every possible way from K₆ reaches exactly seven graphs, the Petersen graph among them, and stops. Every placement of every one of them links, by the same parity count. Remove any edge from any of them, and placements that link nothing appear at once. The seven are the whole answer: a graph must link exactly when it contains one of them.

Four equal quarters with two lines

Any flat shape, however lopsided, can be cut into four pieces of equal area by two perpendicular straight lines. The proof turns the pair of lines like the hands of a clock: the area in one quadrant, minus a quarter, reverses its sign every quarter-turn, so somewhere it is zero. The same kind of argument cuts a solid into eight equal pieces with three planes. It stops working in five dimensions, where some masses cannot be cut into thirty-two equal pieces by five hyperplanes — and in four, nobody knows.

Seven points split three ways

Any seven points in the plane can be divided into three groups whose convex hulls share a point, and six points in general position never can. The theorem is Helge Tverberg's; its topological version, in which straight lines may bend, is true when the number of groups is a power of a prime and false otherwise — and the proof that decides which is the same antipodal argument that halves a sandwich.

Counting targets when the sensors make mistakes

Add up the Euler characteristics of the regions where sensor counts are high and the answer is exactly the number of targets — until one sensor misreads. Each wrong reading is a new piece or a new hole, so the error grows with the number of sensors instead of averaging away. Smooth the readings and count each piece and hole by how long it survives as the threshold is lowered, and the count comes back exactly, with half the readings wrong, at the price of having to know how small a real feature can be.

What it costs to find the point that stays put

Brouwer's theorem promises a point a map leaves where it is, and Sperner's corridor walks to it. On a gentle map the walk is short; twist the map about its fixed point and the corridor follows every turn, so its cost grows like the square of the grid. Halving the square by winding number finds the same point in a number of readings proportional to the grid's side whatever the map does — and no method that only reads the map can do fundamentally better.

Pieces minus holes on a random surface

Cut a random landscape at some height and the region above it breaks into pieces with holes in them. How many pieces, and how many holes, has no formula. Their difference does: the average Euler characteristic of the region above height u is a constant times u multiplied by the bell curve at u, exactly, for every smooth Gaussian surface — because the Euler characteristic is a sum of peaks, pits and saddles, and the average number of those can be computed point by point.

What homology forgets about a loop

Let the letters of a loop commute and the loop group of a space becomes its first homology group: a loop now records only how often it went round each hole. What is thrown away is exactly the loops that bound a surface. On the figure eight that is nearly everything — of the loops of sixty letters that homology calls nought, about one in 6,700 is a loop that actually shrinks.

The room a projective space needs, read off Pascal's triangle

The projective plane cannot sit in three-dimensional space without crossing itself, and the reason can be written as arithmetic: a polynomial that records how a shape twists, which a room must cancel. For the n-dimensional projective space that polynomial is a row of Pascal's triangle read mod 2, its inverse is another row, and the inverse's last term says how many extra dimensions the room must have — exactly enough, at every power of two.

Every loop of the butterfly links every other

The Lorenz equations have infinitely many periodic orbits, each a closed loop winding round the two lobes in its own order. Name each loop by that order — LR, LLR, LLRLR — and the linking number of any two can be read off the names, without solving anything: it is half the number of times a left-going strand passes a right-going one. Every such crossing has the same sign, so every pair of loops links, positively. Computing forty-five pairs from the equations themselves, the Gauss integral lands on the predicted whole number every time.

A knot must turn twice, and a little more

Follow a closed curve and add up how far its direction turns. Any closed curve turns through at least one full circle; a knotted one turns through more than two. The proof is a count of hilltops — the total turning equals 2π times the number of highest points the curve has, averaged over every direction to look from — and a curve with a single hilltop in any direction cannot be knotted.

Torus knots that fill the three-sphere

Hopf filled the three-sphere with circles, every two linked once. Let the circles turn at two different speeds instead — p turns one way while they make q the other — and the three-sphere is filled with (p, q) torus knots, trefoils when p and q are 2 and 3, every two of them linked exactly pq times. Two exceptional fibres remain plain circles; every knot links them q and p times, and they link each other once. Computed from the drawn curves for ten fibrations, every pair comes out at pq to six decimal places.

The pieces no formula counts

A random vibration of a large membrane splits the surface into regions where it bulges up and regions where it dips down. Pieces minus holes has an exact average; the number of pieces has none. Nazarov and Sodin proved it is proportional to the area, Bogomolny and Schmit predicted the constant from percolation, 0.0624, and counting the pieces of 51 random waves gives 0.0599 — below the prediction by several standard errors. The sizes of the pieces follow percolation's law; their number does not.

A cut that gets sharper as the map gets lazier

Iterating a map that shrinks distances by 0.9999 takes 141,254 steps to reach its fixed point to a millionth. Cutting takes six: read the map at one point, and the fixed point must lie on the far side of the line halfway between the point and its image. The cut gets better exactly where iteration gets worse, because a map that barely shrinks puts its fixed point almost on every such line.

An octagon only the hyperbolic plane can hold

Glue an octagon's sides in the pattern that makes a surface with two handles and its eight corners become one point, so their angles must add to a single turn. A flat octagon's add to three. The octagon that fits has corners of 45° and lives in the hyperbolic plane, where its area is forced to be exactly 4π and its four gluing moves are motions of the plane whose commutators multiply to the identity.

A curve of pixels needs two kinds of touching

On a screen a closed curve is a ring of pixels, and whether it has an inside depends on which pixels count as touching. Let diagonal pixels touch, and the inside leaks out through the corners; forbid it, and a curve can fence in a region while being twelve separate dots. Of the 1,048,575 sets of pixels in a small window, every simple closed curve obeys the Jordan curve theorem exactly when the curve and the rest of the plane are given different rules.

An instrument that computes with circles

An astrolabe is a stereographic projection of the sky cut in brass. Because the projection sends circles to circles, every line on it can be drawn with compasses, and turning the star map over the plate is a rotation that answers questions in spherical astronomy without a single calculation. Of the lines medieval makers engraved, one family is not a circle on the sphere — and the arcs they drew for it miss the truth by a third of a millimetre.

How a random path winds round a point

A path in the plane with one point removed has a class in the punctured plane's fundamental group: how many times it has wound round the point. For a random path the class grows like the logarithm of time, and its spread has a law. Brownian motion follows Spitzer's Cauchy law, so heavy-tailed that the mean winding does not exist; a walk on a grid follows the hyperbolic secant law instead, with tails that fall exponentially. The difference is made entirely by close passes, which a grid forbids. With two points removed the class becomes a word, and some loops wind round neither point and still cannot be shrunk.

Which knots a random loop ties

Long random loops are almost always knotted, but knotted as what? Sorting 14,000 closed random polygons by two values of the Alexander polynomial answers it. The trefoil leads at every length, five times as common as the figure-eight at 400 sticks. Each knot's share, divided by the unknotted share, grows like a power of the length, and the trefoil tied twice grows about twice as fast as the trefoil — what a loop carrying independent small tangles would do. Up to 300 sticks the double trefoil turns up exactly as often as two independent trefoils should.

An odd number of squares on every polygon

Does every closed curve that does not cross itself pass through the four corners of some square? Toeplitz asked in 1911, and for polygons and smooth curves the answer is yes, because the squares can be counted and the count is odd. For rectangles of a fixed shape the count is even, which is why they needed a different proof — and for an arbitrary continuous curve the square is still not proved to exist.

The turns a tangent makes

Drive once round a closed track and count how many full turns the steering has made. A circle is one turn, a figure-eight is nought, and every loop adds one. Whitney and Graustein proved that this one number decides when one track can be bent into another without ever stopping — and it can be read off the crossings of a drawing without following the curve at all.

How many ways a curve can cross itself

A closed curve that crosses itself three times can be drawn in exactly six essentially different ways on a sphere, and twenty in the plane. Four crossings give nineteen and eighty-two; six give 376 and 2,645. Every one of them is a word — the order in which the crossings are met — and a choice of direction at each crossing, and the whole census comes from asking which of those choices close up on a sphere.

The lists no covering can have

A covering of the sphere by itself with three branch points is three permutations of the sheets whose product is the identity. The Riemann–Hurwitz count says which lists of cycle shapes might work, and a search through every permutation says which do. Up to thirteen sheets, 150 lists pass the count and fail the search — and not one of them has a prime number of sheets.

Counting the subgroups of a free group

The free group on two letters has one subgroup of index 1, three of index 2, thirteen of index 3, then 71, 461, 3,447 and nearly thirty-two million of index 10. Each is a covering of a figure-eight with a marked sheet, so counting them is counting pairs of permutations that join every sheet to every other. Marshall Hall did it in 1949 by sorting every pair by how far one sheet can travel, and the recursion that comes out says the count is almost exactly n times n factorial.

Lines that leave half on each side

Scatter ten points on a page and draw every line through two of them that leaves four on each side. There are at least five such halving lines and at most thirteen, and every point lies on an odd number of them. Both facts come from turning a line through half a revolution, the argument that halves two shapes with one cut. How many halving lines n points can have is the k-set problem, and its answer lies somewhere between a little more than n and n to the four-thirds.