3 loops in one ring, and the number that separates them
loops is one function. Everything below came out of it during this
build, at parameters taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and if the generator changes, this page
changes with it.
With nothing chosen
Six lists of cycle shapes, and how many coverings each has
The same count as a sum over the characters of S4
4 sheets branched over 3 points of a sphere: a surface of genus 0
3 sheets, 8 of 26 words coming back
The rank of every cover of a wedge of 2 circles
What it checks while it draws
Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.
- rows 0 and 1 of the character table are orthogonal ×42
- on the 1-sheeted cover, the lift of a loop winding 0 times closes exactly when it should ×24
- generator 1 acts as a permutation of the 3 sheets ×14
- point 1's entry is a permutation of the 4 sheets ×13
- edge 1's entry is a permutation of the 3 sheets ×10
- the rank comes out at 1 + 3(2 − 1), which is the index times one less than the rank below ×9
- a loop winding 0 times lifts to a closed path exactly when 0 is a multiple of 3 ×7
- rows 0 and 0 of the character table are normalised ×7
- the class sizes of S2 add to its order ×4
- vertex 1 has at most one a leaving it ×4
- vertex 1 has at most one b leaving it ×4
- face 1 closes up on its own sheet ×3
- the loop drawn for class 0 is a whole number ×3
- 4 sheets, two 4-cycles and a double swap: the character sum and the enumeration agree ×2
- the loop drawn for class -2 is a whole number ×2
- the symmetry returns to the identity first at the 4th power ×2
- 2 sheets, four swaps: the character sum and the enumeration agree ×1
- 3 sheets, three 3-cycles: the character sum and the enumeration agree ×1
- 4 sheets, three double swaps: the character sum and the enumeration agree ×1
- 4 sheets, two double swaps and a 3-cycle: the character sum and the enumeration agree ×1
- a class on the torus is a pair of whole numbers, not both zero ×1
- a free group of rank two has four words of length one ×1
- a spanning tree of a connected graph has one edge fewer than it has vertices ×1
- a straight loop of a coprime class never crosses itself ×1
- a twist is a homeomorphism, so a loop that did not cross itself still does not ×1
- and are not under the other one ×1
- and at most one arriving ×1
- and at the same height ×1
- and by the end the loop has been dragged through the hole ×1
- and consecutive ones differ by exactly one turn ×1
- and every list that is not realised passes the count and the parity anyway ×1
- and exactly one arriving ×1
- and in the second, so it closes up on the torus ×1
- and inside the outer boundary ×1
- and it goes round each hole a net zero times ×1
- and its degree is even ×1
- and its degree is odd ×1
- and on exactly two, once in each direction ×1
- and the answer is a whole number, as a count of lists must be ×1
- and the losses add to an even number, as the sign of a product of permutations forces ×1
- and the same word cancels to nothing on the wedge, where it stays four letters long ×1
- and the top and bottom ×1
- and the top and bottom as many times as its second ×1
- and their number divides the number of sheets ×1
- and they are those multiples in order ×1
- and twelve of length two ×1
- and up ×1
- between one and five words to be refused ×1
- between one and four generator words in a, b and their inverses ×1
- between three and six of the shape lists the family knows are tabulated ×1
- between two and eight branch points ×1
- between two and five sheets ×1
- between two and six sheets ×1
- both loops start at the same point of the ring ×1
- carrying the loop along either path leaves its class alone ×1
- curve is read only by the twist views ×1
- curves is read only by the slopes and simple views ×1
- each loop in the orbit is in its class, across ×1
- each loop on the torus is a pair of whole numbers, not both zero ×1
- each pair: one to four pairs of whole numbers, not both zero ×1
- each shape names a conjugacy class of the group ×1
- each sheet has exactly one edge of each label leaving it ×1
- every branch point's cycles fit side by side inside its column ×1
- every class drawn here is coprime ×1
- every class on the way is coprime ×1
- every coprime class walks down to (1, 0) ×1
- every cover's counted rank matches the formula ×1
- every lifted edge appears on some face ×1
- every marked point actually branches ×1
- every point of the fibre projects onto the marked point of the circle ×1
- every sheet is reached by some word, so the cover is connected ×1
- from is read only by the twistpath view ×1
- going round all the branch points returns every sheet, so this is a covering of the sphere ×1
- going round the face returns every sheet to itself, so the permutations describe a covering ×1
- in the disc every stage of the shrink stays inside ×1
- no sample of the loop lands on the point it is wound about ×1
- no step of the walk turns more than a quarter of a turn ×1
- no two cycles drawn in one lane overlap ×1
- no two deck transformations agree on the base vertex ×1
- one permutation for each of the 2g edges ×1
- one to four pairs of loops ×1
- over the line, the only loop that lifts to a loop is the one that goes nowhere ×1
- pairs is read only by the meet view ×1
- running one loop then the other adds the counts ×1
- so the two maps are not the same map ×1
- so the whole fibre sits over one point ×1
- so there are at most as many symmetries as sheets ×1
- span is read only by the primitive view ×1
- symmetry is read only by the twistorder view ×1
- the a's in the word add up to the winding number about that hole ×1
- the b's in the word add up to the winding number about that hole ×1
- the base circle itself does not lift to a loop ×1
- the bouquet below has two or three circles ×1
- the bouquet has between one and four circles ×1
- the bouquet has two to four circles ×1
- the cells count what the Riemann–Hurwitz formula says ×1
- the character sum and the enumeration agree ×1
- the character table of this symmetric group is one the family carries ×1
- the class is coprime with both counts positive ×1
- the class walked down: one to four pairs of whole numbers, not both zero ×1
- the classes drawn are whole numbers no bigger than four ×1
- the classes that lift to loops are the multiples of the sheet count ×1
- the commutator does not cancel down any further ×1
- the commutator has exponent sum zero in the first generator ×1
- the composition of two deck transformations is one ×1
- the cover has between two and five sheets ×1
- the cover has between two and four sheets ×1
- the cover has genus d(g − 1) + 1 ×1
- the cover is connected, so its deck group acts on one object ×1
- the cover is connected, so the index argument applies ×1
- the cover is connected, so the sheets are one orbit ×1
- the cover's Euler characteristic is even, as a closed orientable surface's must be ×1
- the covering is one of torus, genus2, cuberoot, power, simple ×1
- the curve about the left hole is a whole number ×1
- the curve about the right hole is a whole number ×1
- the curve: one to four pairs of whole numbers, not both zero ×1
- the curves: one to four pairs of whole numbers, not both zero ×1
- the degree of the map that is not odd is a whole number ×1
- the degree of the odd map is a whole number ×1
- the disc lifts to one face on every sheet ×1
- the drawing shows between two and five turns ×1
- the edges outside the tree are exactly the free generators the graph carries ×1
- the Euler characteristic is that of a closed orientable surface ×1
- the Euler characteristic multiplies by the number of sheets ×1
- the first loop is a whole number ×1
- the first map really does send opposite points to opposite points ×1
- the folded graph has at least one independent loop ×1
- the free group outgrows the abelian group of the same rank ×1
- the generator aa is accepted by the folded graph ×1
- the generator ab is accepted by the folded graph ×1
- the generator abab is accepted by the folded graph ×1
- the generator abaB is accepted by the folded graph ×1
- the generator ba is accepted by the folded graph ×1
- the generator bb is accepted by the folded graph ×1
- the hole is between a fifth and three fifths of the ring ×1
- the identity is a deck transformation ×1
- the index of the subgroup is the number of sheets ×1
- the lattice runs from 4 to 12 each way ×1
- the lift ends a whole number of turns above where it started ×1
- the loop at p is a whole number ×1
- the loop being lifted is a whole number ×1
- the loop crosses itself one time fewer than the common factor ×1
- the loop crosses the side edges as many times as its first count ×1
- the loop drawn at p has the class claimed ×1
- the loop drawn for a class really has that class ×1
- the loop is in the class it claims, across ×1
- the loop lifted goes round between one and four times ×1
- the loop lifted is the one asked for ×1
- the loop lifted winds between zero and six times ×1
- the loop stays clear of the hole ×1
- the loops composed have counts no bigger than three ×1
- the loops tabulated wind between zero and eight times ×1
- the marked point is on the base circle ×1
- the monodromy table is drawn over two generators ×1
- the number of sheets is what one lap advances by ×1
- the path crosses the side edges as many times as its first count says ×1
- the permutations connect every sheet to every other ×1
- the permutations reach every sheet from every other, so the covering is connected ×1
- the printed terms add to the printed answer ×1
- the probe's two images are exactly opposite under the odd map ×1
- the second loop is a whole number ×1
- the second map does not, which is what makes it a control ×1
- the shape list is one the family knows ×1
- the share of coprime classes is near 6/π² ×1
- the shrink is legal to begin with and stops being legal ×1
- the slide is drawn in three to five stages ×1
- the surface has genus 1, 2 or 3 ×1
- the symmetries reach every sheet exactly when there are as many of them as sheets ×1
- the symmetry is ab or aba ×1
- the table holds a list that is realised and a list that is not ×1
- the table runs to between three and six sheets ×1
- the table runs to between two and six sheets ×1
- the twist is done one to three times ×1
- the twisted loop crosses the side edges as the matrix predicts ×1
- the two run one after the other is a whole number ×1
- the two squares never overlap during the slide ×1
- the view is one the family draws ×1
- the walk ends ×1
- the winding number is a whole number ×1
- the word a is refused ×1
- the word aa is refused ×1
- the word ab is refused ×1
- the word b is refused ×1
- the word is one this family realises ×1
- the word read off the curve is the word the curve was built from ×1
- the words that come back are closed under composition ×1
- times is read only by the twist view ×1
- two or three circles below ×1
- two straight loops meet as often as the determinant of their classes says ×1
- two words land on the same sheet exactly when one undoes the other into the subgroup ×1
- while a word that really is trivial reduces away, so the test can fail ×1
- with the last point's permutation removed, going round no longer returns every sheet ×1
- words of length up to two to four are listed ×1
Where it is called
Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.
A ball whose outside is not one
Alexander's sphere separates space into two pieces, exactly as the theorem promises. Its inside is an ordinary ball. Its outside is not, and the obstruction is a tree of clasped horns whose tips never stop.
TopologyA count that can say zero
The branched count ends on a list of cycle shapes that passes every test and describes no covering. There is an exact formula for how many coverings a list has — a sum over the character table of a symmetric group — and it returns nought without giving any reason why.
TopologyA covering is a permutation
Describing a covering means saying where each loop sends each sheet, which is a permutation for every generator. So a covering of a wedge of circles is nothing but a homomorphism to a symmetric group, and the subgroup it corresponds to is a stabiliser.
TopologyA loop that cannot be pulled tight
A hole is a strange thing to point at, because it is precisely where the surface is not. What can be pointed at is a loop of string lying on the surface — and the hole announces itself by refusing to let that loop be pulled in to a point.
TopologyA 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.
TopologyCovering a surface multiplies its count
A covering of a closed surface is a permutation of the sheets for each edge of the surface's one face — with one condition that a covering of a graph never had to meet. When the condition holds, the cells of the cover can be counted directly, and the count is the base's count times the number of sheets. That multiplication decides which surfaces can cover which, before any cover is built.
TopologyCutting a space to find its group
A space assembled from two pieces has a fundamental group assembled from theirs, and the recipe is exact — take everything both groups offer and impose the relations the overlap forces. Almost every fundamental group anybody knows is computed this way, including all of the surfaces.
TopologyEvery cover is a subgroup
A space can be unrolled, and the ways of unrolling it are not arbitrary. They correspond exactly to the subgroups of its fundamental group — index equals sheets, normality equals symmetry — so a question about a group becomes a question about a picture and back again.
TopologyFolding a graph until it decides
A subgroup of a free group usually arrives as a list of words, and almost nothing about it is readable from the list. Draw the words as loops, merge every pair of edges with the same label leaving one point, and what is left is a machine that decides membership by reading.
TopologyLinked, and no two of them are
Three rings that cannot be pulled apart, in which every pair comes apart the moment the third is removed. Every pairwise linking number is zero, so the number cannot see it — and what does see it is a word in two letters that refuses to cancel.
TopologyOne line that halves them both
Two shapes lying anywhere on a page, of any sizes and any shapes at all. There is always a single straight line that cuts both of them into two equal halves at once — and finding it needs no cleverness, only the observation that a quantity which reverses sign has to pass through zero.
TopologyThe group a space has at a point
The loops of a space form a group once a starting point is fixed, and the fixing looks like an arbitrary choice that ought to be removable. It is removable, but only up to conjugation, and the residue is exactly what makes a non-commutative fundamental group harder to state than to compute.
TopologyThe 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.
TopologyThe same loop, unrolled
Spread a circle out into a line spiralling above it, and a loop that closes downstairs becomes a path that does not — so a question about which loops can be shrunk becomes a question about where a path ends, which is easy.
TopologyThe subgroup that is freer than the group
A free group on two letters contains a subgroup of index three that is free on four. Nothing about a group makes that plausible; everything about a graph makes it obvious, and the argument is to stop looking at the group and start looking at the space whose loops it is.
TopologyThe symmetries a cover has of its own
A covering space can be shuffled without disturbing anything below it, and how many ways there are is decided by the subgroup it corresponds to. When there are as many symmetries as sheets the covering is called regular, and that is the same statement as the subgroup being normal.
TopologyTwo loops and one number
Give each crossing between two closed curves a sign, add them up, halve — and the answer does not depend on how the curves were drawn, how they are pushed about, or which way the picture was projected.
TopologyWhat a branch point subtracts
Let the sheets of a covering meet at a few points and the count stops multiplying — but it fails by an amount that can be read off each point's permutation. Cut the sphere into a star, lift the cells, and the Riemann–Hurwitz formula falls out of a subtraction. The same count then turns out to be necessary and not sufficient.
TopologyWhere the fixed point escapes
The theorem asks for a set that is closed, bounded and free of holes. Drop any one of the three and a map appears that moves every single point — and in each case the point that should have stayed still can be seen leaving.
TopologyWhy the second group commutes
Replace loops by spheres and the same construction gives a second homotopy group. It is always commutative, and the reason is not a fact about spheres or about any space — it is a two-line argument about any set carrying two compatible operations.