Generator

the trefoil

A generator in the topology library, called 128 times across 23 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

knot 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

the trefoil. the trefoil, drawn as a closed curve with 3 crossings. At each crossing the strand passing underneath is broken, which is the only information the flat picture carries that the curve alone does not.

A trefoil, and how far its tangent turns

In every direction the trefoil has at least two highest points

Total curvature is a count of hilltops, averaged over directions

Squeezing a trefoil towards 4π

Where the tangent points: the circle's path and the trefoil's

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.

linking of fibres 0 and 1 ×21
3_1 is not the unknot ×14
3_1's braid gives the determinant in the tables ×14
both checkerboard colourings give 3_1 changed's signature ×14
both checkerboard colourings give 7_4's signature ×14
the changes found for 3_1 are the tables' unknotting number ×14
both checkerboard colourings give T(2,7)'s signature ×4
in every placement of G7 the linking numbers over disjoint cycle pairs add to an odd number ×3
in G7 every two disjoint edges are separated by an even number of cycle pairs, so a crossing change keeps the parity ×3
the Borromean rings: the diagram's count and Gauss's integral agree for components 1 and 2 ×3
the figure-eight knot: 3 colours are available exactly when 3 divides its determinant ×3
the figure-eight knot: the count with 3 colours ×3
the trefoil: 3 colours are available exactly when 3 divides its determinant ×3
the trefoil: the count with 3 colours ×3
the unknot: 3 colours are available exactly when 3 divides its determinant ×3
the unknot: the count with 3 colours ×3
three rings in a chain: the diagram's count and Gauss's integral agree for components 1 and 2 ×3
2π times the average number of maxima is the total curvature ×1
a colouring is drawn on a diagram that has crossings ×1
a colouring uses three, five or seven colours ×1
a crossing change moves the signature by at most two ×1
a crossing joins two different Seifert circles ×1
a diagram with k crossings is cut into k arcs ×1
a knot and its mirror image have the same Alexander polynomial ×1
a knot's Goeritz form has no zero eigenvalue ×1
a knot's signature is even ×1
a loop that dips through and back: punctures and crossings agree ×1
a loop that dips through and back: the diagram's count and Gauss's integral agree for components 1 and 2 ×1
a pentagon with five crossings in its picture is found ×1
a polygon of each kind is found among the first four thousand ×1
a regular fibre links the exceptional circles q and p times, and they link once ×1
a seven-stick polygon is an unknot, a trefoil or a figure-eight ×1
a switched crossing is one the diagram has, named once ×1
a torus knot squeezed onto its core turns 2π per circuit of the axis ×1
almost every position along the path is in general position ×1
an unknot with one twist has the crossings it claims ×1
an unknot with one twist: an alternating diagram with a crossing that untwists has a bracket narrower than 4c ×1
an unknot with one twist: an alternating diagram's extreme states meet themselves exactly at its untwistable crossings ×1
an unknot with one twist: an alternating diagram's two extreme states have c + 2 circles between them ×1
an unknot with one twist: the bracket gives the Jones polynomial the handedness requires ×1
and a Lissajous curve with each of their polynomials is in the census ×1
and Gauss's integral, which sees no disc at all, gives it too ×1
and it is unknotted ×1
and its determinant ×1
and its determinant is 1 or 7 modulo 8 ×1
and one whose linking number is bigger than one ×1
and the one found uses all three ×1
and their signs add to an even number ×1
and they reach zero in two different ways — one with no punctures, one with two that cancel ×1
at every crossing the three arcs are all alike or all different ×1
at least two of the links have linking number zero ×1
between −3 and 4 extra full turns ×1
between three and five deformation sizes, none of them large ×1
between two and four links the family knows ×1
Călugăreanu: the linking number is twist plus writhe ×1
changing a negative crossing never raises it ×1
changing a positive crossing never lowers the signature ×1
Conway and Gordon: the a₂ of the 360 cycles add to an odd number ×1
Conway and Gordon: the ten linking numbers add to an odd number ×1
count is read only by the stickears, stickcensus, randdets, k7many, k6knots views ×1
crossings grow like n for walks and n² in a box; the mean square linking number like n² in a box and far more slowly than the crossings for walks ×1
each crossing has exactly one of its two strands going underneath ×1
each drawn polygon's full polynomial agrees with its identification ×1
every curve stays above 4π ×1
every disjoint pair in the Petersen graph is two pentagons ×1
every edge survives trimming at both ends ×1
every embedding has an odd sum of linking numbers ×1
every five-pointed star a pentagon makes has twists three one way and two the other ×1
every knot's determinant is odd ×1
every knotted curve exceeds 4π ×1
every Lissajous knot's polynomial is a square modulo 2 ×1
every one has fifteen edges ×1
every orbit's genus is a whole number ×1
every pair of orbits up to length 8 links at least once ×1
every pair of trefoil fibres links six times ×1
every placement has a trefoil among its paths ×1
every placement has an odd number of knotted paths ×1
every polygon of seven or fewer sticks has determinant 1, 3 or 5 ×1
every power of A in the normalised bracket is a multiple of four, as a knot's must be ×1
every row of the matrix vanishes at t = 1, as a presentation of a knot must ×1
every state leaves at least one circle ×1
few enough circles to try every ordering ×1
following the smoothing from a segment comes back to that segment ×1
from 100 to 300 sticks the double trefoil comes within a fifth of the rule ×1
Gauss's integral lands on a whole number ×1
in every direction the knotted curve has at least two maxima ×1
in every placement of K₃,₃,₁ the linking numbers over disjoint cycle pairs add to an odd number ×1
in every placement of K₄,₄ minus an edge the linking numbers over disjoint cycle pairs add to an odd number ×1
in every placement of K₆ the linking numbers over disjoint cycle pairs add to an odd number ×1
in every placement of Petersen graph the linking numbers over disjoint cycle pairs add to an odd number ×1
in K₃,₃,₁ every two disjoint edges are separated by an even number of cycle pairs, so a crossing change keeps the parity ×1
in K₄,₄ minus an edge every two disjoint edges are separated by an even number of cycle pairs, so a crossing change keeps the parity ×1
in K₆ every two disjoint edges are separated by an even number of cycle pairs, so a crossing change keeps the parity ×1
in Petersen graph every two disjoint edges are separated by an even number of cycle pairs, so a crossing change keeps the parity ×1
its determinant is 3, so it is knotted ×1
its determinant is 5 ×1
its polynomial is the figure-eight's, −t + 3 − t⁻¹, from four directions ×1
its polynomial is the trefoil's, t² − t + 1, from four directions ×1
knots sharing a determinant are told apart by the polynomial ×1
Levine: four divides the signature exactly when the determinant is 1 mod 4 ×1
longer polygons are knotted far more often ×1
loops far apart are rarely linked ×1
Milnor's identity holds for every curve ×1
most placements of six points have no knotted path ×1
no hexagon is a figure-eight ×1
no knot is undone with fewer changes than its signature allows ×1
no orbit of period four or less is knotted ×1
no pentagon had two corners' triangles pierced ×1
no pentagon is knotted ×1
no single arc passes over every crossing ×1
one boundary circle, so the genus is (1 − χ)/2 ×1
one crossing left is the unknot; three is not ×1
reflection negates the signature ×1
seen from above it has 2ab − a − b crossings ×1
signed crossings between two trefoil fibres ×1
six points have 60 closed paths through all of them ×1
six sticks make only unknots and trefoils ×1
so the number of knotted paths is odd too ×1
so the number of linked pairs is odd too, since every linking number here is 0 or ±1 ×1
some number of extra turns keeps the linking number ×1
squeezed onto the core, the curvature nears 2π per turn about the axis ×1
sticks is read only by the randone, randclose views ×1
the (2, 4) torus link: the diagram's count and Gauss's integral agree for components 1 and 2 ×1
the (2, c) torus knot has signature 1 − c ×1
the a₂ sum is odd ×1
the a₂ sum is odd for every placement ×1
the a₂ sum stays odd as the point moves ×1
the average of the directional counts is the writhe ×1
the bracket lies inside the window its two extreme states allow ×1
the bracket's span is at most four times the crossings ×1
the cinquefoil admits a colouring using more than one of the 5 colours ×1
the cinquefoil has the crossings it claims ×1
the cinquefoil takes ± its determinant at θ = π ×1
the cinquefoil takes the value 1 at θ = 0 ×1
the cinquefoil, one crossing switched: a diagram that does not alternate has a bracket narrower than 4c ×1
the cinquefoil: |Δ(−1)| is the determinant ×1
the cinquefoil: a reduced alternating diagram's bracket spans exactly 4c ×1
the cinquefoil: an alternating diagram's extreme states meet themselves exactly at its untwistable crossings ×1
the cinquefoil: an alternating diagram's two extreme states have c + 2 circles between them ×1
the cinquefoil: on an alternating diagram the two bounds agree ×1
the cinquefoil: the bracket gives the Jones polynomial the handedness requires ×1
the classification is checked on at least a hundred and fifty small diagrams ×1
the composite's slope is about twice the trefoil's ×1
the composites' share rises with length ×1
the constant colourings are always allowed, so there are at least p ×1
the core curve is one the family knows ×1
the corrected steps return to the start ×1
the crossing count and Gauss's integral agree ×1
the curve is the knot 5₂ (polynomial 2t − 3 + 2t⁻¹) ×1
the curve stays clear of the projection point ×1
the decay length is a few hundred sticks ×1
the discs and bands form one connected surface ×1
the drawn view is the same knot ×1
the family contains the Petersen graph, K₃,₃,₁ and K₄,₄ minus an edge ×1
the figure-eight knot admits a colouring using more than one of the 5 colours ×1
the figure-eight knot has the crossings it claims ×1
the figure-eight knot takes ± its determinant at θ = π ×1
the figure-eight knot takes the value 1 at θ = 0 ×1
the figure-eight knot, one crossing switched: a diagram that does not alternate has a bracket narrower than 4c ×1
the figure-eight knot: |Δ(−1)| is the determinant ×1
the figure-eight knot: a reduced alternating diagram's bracket spans exactly 4c ×1
the figure-eight knot: an alternating diagram's extreme states meet themselves exactly at its untwistable crossings ×1
the figure-eight knot: an alternating diagram's two extreme states have c + 2 circles between them ×1
the figure-eight knot: on an alternating diagram the two bounds agree ×1
the figure-eight knot: the bracket gives the Jones polynomial the handedness requires ×1
the first component of this link is a flat circle, so its disc is the obvious one ×1
the five twists are three one way and two the other ×1
the four ends at a crossing alternate over, under, over, under ×1
the Gauss integral of the computed orbits agrees with the template ×1
the heptagon: the polynomial is the same from every direction ×1
the hexagon: the polynomial is the same from every direction ×1
the Hopf link: punctures and crossings agree ×1
the Hopf link: the diagram's count and Gauss's integral agree for components 1 and 2 ×1
the Jones polynomial of a knot takes the value 1 at t = 1 ×1
the knot is one of 3_1, 4_1, 5_1, 5_2, 6_1, 6_2, 6_3, 7_1, 7_2, 7_3, 7_4, 7_5, 7_6, 7_7 ×1
the knot is one the family draws, with crossings ×1
the knot is one the figure knows ×1
the knot's Jones polynomial is one the family knows ×1
the knots in the table have different determinants ×1
the knotted curve turns through more than two full turns ×1
the link is one the family knows ×1
the linking number has heavier tails than a bell ×1
the linking number is the same at every deformation ×1
the minor is not zero ×1
the mirror image of the trefoil has the crossings it claims ×1
the mirror image of the trefoil: a reduced alternating diagram's bracket spans exactly 4c ×1
the mirror image of the trefoil: an alternating diagram's extreme states meet themselves exactly at its untwistable crossings ×1
the mirror image of the trefoil: an alternating diagram's two extreme states have c + 2 circles between them ×1
the mirror image of the trefoil: the bracket gives the Jones polynomial the handedness requires ×1
the mirror's Jones polynomial is the original's with t replaced by 1/t ×1
the modular computation agrees with the symbolic one ×1
the modular determinant is the exact one ×1
the number of embeddings is a whole number between 200 and 5000 ×1
the number of linked pairs changes by an even amount ×1
the orbit closes after one period ×1
the orbit winds round the lobes in the order its name says ×1
the orbits are among LR, LLR, LRR, LLLR, LLRR, LRRR, LLLRR, LLRLR, LLRRR, LRLRR ×1
the pentagon: the polynomial is the same from every direction ×1
the picture from above has three crossings ×1
the polygon formula gives the integer once the polygons are fine enough to stay disjoint ×1
the polynomial at t = −1 is the determinant found from the colourings ×1
the polynomial is symmetric ×1
the polynomial is symmetric — the same read backwards ×1
the polynomial takes the value ±1 at t = 1 ×1
the polynomial's bound never exceeds the surface's genus ×1
the punctures added with signs give the linking number ×1
the ribbon closes up ×1
the seed is a whole number between 1 and 100000 ×1
the seven-crossing torus knot has the crossings it claims ×1
the seven-crossing torus knot, one crossing switched: a diagram that does not alternate has a bracket narrower than 4c ×1
the seven-crossing torus knot: |Δ(−1)| is the determinant ×1
the seven-crossing torus knot: a reduced alternating diagram's bracket spans exactly 4c ×1
the seven-crossing torus knot: an alternating diagram's extreme states meet themselves exactly at its untwistable crossings ×1
the seven-crossing torus knot: an alternating diagram's two extreme states have c + 2 circles between them ×1
the seven-crossing torus knot: on an alternating diagram the two bounds agree ×1
the seven-crossing torus knot: the bracket gives the Jones polynomial the handedness requires ×1
the signed crossings add to twice the linking number ×1
the signed crossings of two closed curves add to an even number ×1
the strand is broken where it goes underneath, not somewhere else ×1
the sum stays odd at every position ×1
the surface's Euler characteristic gives a whole genus ×1
the table contains a link whose components cross and whose linking numbers are all zero ×1
the table is ordered by crossing number ×1
the table's columns are small primes ×1
the table's diagrams alternate until a crossing is switched ×1
the total is odd in this placement too ×1
the trefoil admits a colouring in more than one colour ×1
the trefoil admits a colouring using more than one of the 3 colours ×1
the trefoil has the crossings it claims ×1
the trefoil is the commonest knot at every length from 100 ×1
the trefoil is three arcs ×1
the trefoil takes ± its determinant at θ = π ×1
the trefoil takes the value 1 at θ = 0 ×1
the trefoil, one crossing switched: a diagram that does not alternate has a bracket narrower than 4c ×1
the trefoil: |Δ(−1)| is the determinant ×1
the trefoil: a reduced alternating diagram's bracket spans exactly 4c ×1
the trefoil: an alternating diagram's extreme states meet themselves exactly at its untwistable crossings ×1
the trefoil: an alternating diagram's two extreme states have c + 2 circles between them ×1
the trefoil: on an alternating diagram the two bounds agree ×1
the trefoil: the bracket gives the Jones polynomial the handedness requires ×1
the trefoil's share of knotted loops falls with length ×1
the trefoil's share rises with length to 400 sticks ×1
the tube radius is between 0.05 and 0.9 ×1
the two computations agree at this deformation ×1
the two loops stay clear of each other at every amplitude ×1
the two polynomial values identify the knot the full polynomial names ×1
the two strands at a crossing are not parallel ×1
the unknot is a single arc ×1
the view is one the family draws ×1
there are nine three-colourings of the trefoil in all ×1
there are three moves, and Reidemeister proved there are no others ×1
triangle–star exchanges from K₆ reach exactly seven graphs ×1
turning the third phase alternates between 5₂ and the unknot ×1
twice the over-strand equals the two under-strands added, at every crossing ×1
twist and writhe add to the held integer at every shape ×1
two circles side by side: punctures and crossings agree ×1
two circles side by side: the diagram's count and Gauss's integral agree for components 1 and 2 ×1
two closed curves cross an even number of times ×1
two edges never meet in space ×1
two independent runs agree on the mean square linking number ×1
two or three links whose first component is a flat circle ×1
two projections agree after the shift ×1
two projections give the same linking number ×1
two projections give the same polynomial ×1
two projections give the same whole number ×1
two regular fibres link pq times ×1
two triangles' signed crossings add to an even number ×1
two words in L and R of length 2 to 8, each using both letters ×1
up to seven crossings exactly 5₂, 6₁, 7₄ and 7₅ pass both tests ×1
while the number of crossings is not ×1
with one edge deleted, some random placement links nothing ×1
x is unchanged ×1
y changes sign ×1
z changes sign ×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.

With nothing chosen

A trefoil, and how far its tangent turns

In every direction the trefoil has at least two highest points

Total curvature is a count of hilltops, averaged over directions

Squeezing a trefoil towards 4π

Where the tangent points: the circle's path and the trefoil's

What it checks while it draws

Where it is called

A knot must turn twice, and a little more

A polynomial behind the colourings

A polynomial that tells left from right

A whole number split into two that are not

Almost every long loop is knotted

Colours that count more than three

Every loop of the butterfly links every other

How many changes undo a knot

Linked, and no two of them are

Many crossings and a small linking number

Seven graphs that must link

Seven points and a knot they cannot avoid

Six points in space and a pair that must link

Six sticks tie a trefoil, and five cannot

The crossings an alternating knot cannot lose

The surface a knot bounds

Three moves, and what they cannot undo

Three sines tie a knot

Torus knots that fill the three-sphere

Two loops and one number

Two pieces, in every dimension

Which knots a random loop ties

Zero can mean two different things