Field

Topology — page 3

What survives bending, and what does not.

The Seifert circles of the trefoil. The trefoil with an orientation, cut at each of its 3 crossings and reconnected the way the orientation allows. The 6 segments form 2 circles, and the surface built from them has genus 1.

The surface a knot bounds

Every knot is the edge of a surface with two sides, and Seifert found a way to build one from any diagram: smooth the crossings, fill the circles that result with discs, and join them with twisted bands. Counting the handles gives an upper bound on how complicated the knot is, the Alexander polynomial gives a lower one, and for the simplest knots the two meet.

Every way to pair the edges of a hexagon. Chord diagrams of all 15 pairings of a 6-gon's edges, shaded by the surface each gluing makes: 5 spheres, 10 tori.

Every way to pair a polygon's edges

A hexagon's six edges can be paired in fifteen ways. Glue each pair head to tail and five of the fifteen give a sphere and ten give a torus; an octagon's 105 pairings give 14 spheres, 70 tori and 21 surfaces with two handles. The spheres are exactly the pairings whose chords never cross, and the whole table obeys one recurrence found in 1986.

A random pairing of 24 edges. Chord diagram of one uniformly random pairing of a 24-gon's edges, corners coloured by the vertex they become. a 24-gon with its edges paired at random and glued head to tail: the 24 corners fall into 3 vertices, so the surface has genus 5, against a most possible of 6.

The surface a random gluing makes

Pair the edges of a large polygon at random and glue each pair head to tail. The surface almost always has nearly as many handles as the polygon allows: a thousand edges leave about seven and a half vertices, and the genus is within four of its ceiling of 250. The vertices behave like the cycles of a random permutation, and their average is a harmonic number.

A labelled square and the edges that join opposite labels. A square grid of 121 vertices, each coloured by one of four labels, with opposite boundary vertices carrying opposite labels, and 3 edges drawn thick where a label meets its negative.

Opposite labels that have to meet

Cut a square into triangles, label every corner +1, −1, +2 or −2, and insist only that opposite points of the edge get opposite labels. Somewhere inside, an edge must join a label to its negative. The proof counts quarter-turns round a diamond — an odd number on the boundary, zero in any triangle that avoids opposites — and making the triangles smaller turns the count back into the theorem about opposite points on the Earth.

The pairs from five, joined when disjoint, need three colours. The Petersen graph drawn with its ten vertices labelled by pairs from one to five, edges joining disjoint pairs, and a proper colouring with three colours.

The colours a circle forces

Take every pair from five things and join two pairs when they share nothing. Three colours are enough to colour the result so joined pairs differ, and two are not — but no triangle, no dense cluster and no counting argument explains why. The reason is five points on a circle and a direction that cannot be told apart from its opposite, and the same reason, one sphere at a time, settles Kneser's question for every size.

Twelve units of shortfall, on every solid with three faces at a corner. A bar for each of 8 polyhedra with three faces at every vertex, divided into each face's shortfall from six sides; every bar has total length twelve, and the hexagons contribute nothing.

Twelve pentagons, whatever the hexagons

A football has twelve pentagons and twenty hexagons. A molecule of sixty carbon atoms has the same pattern, a molecule of seventy has twelve pentagons and twenty-five hexagons, and a geodesic dome of any size has twelve places where the pattern of six breaks. None of this is a coincidence of design: Euler's formula, rearranged, says that faces meeting three at a corner must fall short of hexagons by exactly twelve in total, and the hexagons are free.

The six regular 4-polytopes, and an alternating sum of 0. A table of the six regular polytopes in four dimensions with their numbers of vertices, edges, faces and cells and the alternating sum, which is zero for each.

Zero in four dimensions

Corners minus edges plus faces is two for every solid. One dimension up, corners minus edges plus faces minus cells is zero for every one of the six regular four-dimensional solids, from the five-cell to the six-hundred-cell, and for every other convex solid in four dimensions. The alternating sum does not break when the dimension rises: it alternates, two in odd dimensions and zero in even ones, because it is measuring a sphere and not a solid.

Loops on a torus that never cross themselves. Squares with opposite edges glued, each carrying one straight loop of a different slope, each labelled with its two crossing counts.

The loops on a torus that never cross themselves

Every loop on a torus is classified by two whole numbers: how often it goes round one way and how often the other. Some classes can be drawn without the loop ever crossing itself and some cannot, and the rule is the oldest in arithmetic — the two numbers must have no common factor. The same two numbers say how often any two loops must meet.

A twist along the horizontal loop, done once. Squares with opposite edges glued and a shaded horizontal band, showing one loop before and after the torus is twisted along the band.

A twist that carries one loop to another

Cut a torus along a loop, turn one side of the cut once round, and glue it back. Nothing is torn, so every loop that did not cross itself still does not — but a loop of class (0, 1) is now a loop of class (1, 1). Two such twists reach every loop that never crosses itself, by Euclid's algorithm, and the symmetries they generate are exactly the whole-number matrices of determinant one.

The trefoil and its mirror image. The trefoil beside its reflection, with writhe 3 and −3. The Jones polynomials are t + t³ − t⁴ and −t⁻⁴ + t⁻³ + t⁻¹; the Alexander polynomial of both is t − 1 + t⁻¹.

A polynomial that tells left from right

The trefoil and its mirror image have the same colourings, the same determinant and the same Alexander polynomial, and the first proof that they differ was a hard argument about groups. Smooth every crossing both ways, count the circles in each of the resulting pictures, and add up the counts with the right weights: the total changes when the knot is reflected.

The two extreme states of the figure-eight knot. The figure-eight knot, then the state with every crossing smoothed the A way (3 circles) and the state with every crossing smoothed the B way (3 circles). The bracket spans 16 powers of A against 16 for four times the crossings.

The crossings an alternating knot cannot lose

Peter Guthrie Tait drew knots for years and believed, without proof, that a diagram whose crossings alternate over and under, and which has no twist that can be undone, is already drawn with the fewest crossings the knot allows. The proof took a century, and when it came it needed only the two most extreme ways of smoothing the diagram and Euler's count of the regions of a map.

How many changes undo a knot

Cut the string at a crossing, pass it through the other strand and join it up again, and any knot can be undone by doing that often enough. The fewest changes needed is the unknotting number, and proving that fewer will not do needs a number that each change can move only a little. The signature moves by at most two per change — enough to settle thirteen of the fourteen knots up to seven crossings, and not the fourteenth.

Several colours on every vertex

Give every pair from six points three colours, so that pairs with nothing in common share no colour. Counting says nine colours might do; ten are needed. Stahl conjectured in 1976 exactly how many colours every such problem needs — a formula that meets Lovász's topological answer at one colour a vertex and the obvious answer at k — and a search over stars and triangles confirms it in every case small enough to run.

A ribbon on a lifted figure eight: link 1, twist 0.47, writhe 0.53. A closed ribbon drawn as its core curve and one edge, joined by short ties. The edges have linking number 1; the twist 0.467 and writhe 0.533 add to it.

A whole number split into two that are not

Run a ribbon round a closed loop and its two edges link a whole number of times. That number is shared between two quantities that are nothing like whole numbers: how far the ribbon twists about its core, and how far the core coils about itself. Bend the loop and the twist and the coiling trade continuously, to three decimal places, while their sum stays fixed — the arithmetic behind a coiled telephone cord and a supercoiled loop of DNA.

Six points in space and the triangles that link, seed 48. A projection of K₆ with straight edges, the under-strands broken at crossings. 1 of the ten pairs of disjoint triangles are linked; the pair 126 and 345 is coloured.

Six points in space and a pair that must link

Put six points anywhere in space and join every pair with a straight segment. Split the six into two triangles — there are ten ways — and at least one of the ten pairs of triangles is linked like two rings of a chain. No placement avoids it. The reason is a parity: moving an edge through another changes exactly two of the ten linking numbers, so their sum stays odd whatever is done.

Every count of corners and faces a convex solid can have. A grid of pairs (V, F) from 4 to 16, with the 85 realisable pairs filled between the lines F = 2V − 4 and V = 2F − 4, and the regular solids labelled.

Every count a solid can have

Euler's formula ties the corners, edges and faces of a convex solid together, but it does not say which numbers are possible. Ernst Steinitz answered in 1906: V corners and F faces occur together exactly when neither is more than twice the other less four. The proof in one direction is two lines of counting; in the other it is two moves — cut off a corner, glue on a tetrahedron — that walk from the pyramids to every allowed pair.

Counting overlapping targets from sensor counts alone, by integrating against χ. A 36 by 32 grid of sensor counts from 7 overlapping discs; the sum of the Euler characteristics of the level sets is 7.

Counting targets by their holes

The Euler characteristic adds up like an area: glue two shapes together and it is the sum of theirs minus that of their overlap. So it can be used to measure, and measuring with it does something no area can. A field of sensors that each report only how many targets they detect — not which, not where — can have its readings added up, weighted by the Euler characteristics of the regions where the count is high, and the answer is exactly the number of targets.

Every way to cut a genus-2 surface into pairs of pants. 2 thickened graphs, one for each type of pants decomposition of the closed surface of genus 2, each with 2 coloured junctions and 3 cutting circles.

Every surface is sewn from pants

A sphere with three holes — a pair of pants — is the smallest piece a surface with two or more handles can be cut into. Every such surface falls apart into them, and the Euler characteristic alone says how many pieces and how many cuts, however the cutting is done. What it does not say is the pattern, and the patterns are counted by drawing each one as a graph: two for two handles, five for three, seventeen for four.

The seven-vertex torus, unrolled onto a lattice. A triangular lattice with every point labelled a + 3b mod 7 and fourteen triangles shaded as one copy of the torus.

The fewest corners a surface needs

Build a closed surface from triangles, any two meeting along a whole edge, at a single corner or not at all, and ask for the fewest corners. Two lines of counting give a floor for every surface, in terms of its Euler characteristic alone. The torus meets it with seven, the projective plane with six — and the Klein bottle, which the counting says could be built from seven, cannot, as a search of every possible arrangement shows.

The two-sided cover of a sphere with 3 cross-caps, built from aabbcc. Two copies of the polygon aabbcc with opposite senses of turning, glued within or across copies; 2 vertices, 6 edges, 2 faces, characteristic −2: a surface with 2 handles.

Two copies of the polygon, cross-matched

The two-sided surface that lies over a one-sided one can be built from the gluing word alone: take two copies of the polygon, one read each way round, glue a letter within the copies when it is used once each way, and across them when it is used twice the same way. The recipe works for every surface at once, doubles every count, and shows that a sphere with k cross-caps is covered by the surface with k − 1 handles — and that among all its connected double covers, exactly one is two-sided.

Every section of the Möbius band crosses the zero section. The Möbius band drawn flat, with 3 continuous sections; zeros per section: 1, 3, 1.

Every section of the band must vanish

Read the Möbius band as a line standing over each point of a circle, and choose a point on each line continuously: a section. On a cylinder a section can stay away from zero all the way round. On the band it cannot — every section crosses zero an odd number of times — and that single fact is what orientability means for a bundle. There are exactly two line bundles over a circle, told apart by one sign, and two Möbius bands added together make the plain cylinder's twin.

A trefoil knot made of six straight sticks. Hexagon with corners (6, 2, 3), (7, 7, 10), (1, 6, 4), (2, 1, 7), (8, 10, 5), (0, 9, 9); three crossings from above, determinant 3, Alexander polynomial t² − t + 1.

Six sticks tie a trefoil, and five cannot

Build a knot from straight sticks joined end to end and ask for the fewest. A trefoil takes six, and the six corners can be whole-number points in a box ten units wide. Five sticks can cross one another five times in a picture, as often as a cinquefoil needs, and still tie nothing — the five crossings always twist three one way and two the other. The fewest sticks is a measure of how knotted a knot is that no diagram shows directly.

How often a closed random polygon is certainly knotted, by length. 10: 0.8% (mean crossings 2.8); 20: 1.5% (mean crossings 7.9); 40: 6.3% (mean crossings 21.2); 60: 10.5% (mean crossings 35.1); 80: 20.8% (mean crossings 51.3); 100: 22.0% (mean crossings 66.9); 130: 32.3% (mean crossings 94.9); 160: 45.7% (mean crossings 122.6); 200: 47.0% (mean crossings 154.5); 250: 51.7% (mean crossings 205.5).

Almost every long loop is knotted

Close a random walk into a loop and ask whether it is knotted. With ten steps almost never; with a hundred, more than one time in five it can be proved knotted by a single number; with two hundred and fifty, more than half. The chance of staying unknotted falls exponentially with length — Frisch, Wasserman and Delbrück guessed it for polymer rings around 1961, and it was proved in 1988 — because a knot needs only one small tangle somewhere, and a long loop has room for many.

Seven points in space and a knotted path through all of them, seed 12. A projection of K₇ with straight edges, faint, and one Hamiltonian cycle drawn heavy with its crossings broken: a trefoil. 9 of the 360 Hamiltonian cycles are knotted.

Seven points and a knot they cannot avoid

Put seven points anywhere in space and join every pair with a straight segment. There are 360 closed paths that visit all seven points once each, and at least one of them is knotted — however the points are placed. The reason is a parity, as it was for six points and a linked pair: a number read off each path's knot adds up, over all 360, to something odd. Six points are not enough; seven always are.

All essays