The collection

Every essay — page 20

Page 20 of 46, continuing through the fields in the same order.

Geometry Analysis Algebra Discrete Topology Probability Number Dynamics Logic Computation Applied What's new Series Concepts Search

Topology

What survives bending, and what does not.

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

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 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

How many changes undo a knot

Several colours on every vertex of a Kneser graph. A grid of Kneser graphs on pairs from five to eight points against the number of colours per vertex, each cell giving the fewest colours needed and the counting lower bound.

Several colours on every vertex

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

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

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

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

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

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

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

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

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

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

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

Seven graphs that must link

Four equal quarters of an L-shaped plate. An L-shaped plate cut by two perpendicular halving lines at 48.8 degrees into four pieces of equal area.

Four equal quarters with two lines

Seven points split three ways

Counting targets from sensor readings with mistakes in them. Three 80 by 72 sensor fields over 8 targets: true counts (integral 8), counts with 415 mistaken readings (integral 160), and the mistaken counts smoothed (rounded persistence count 8).

Counting targets when the sensors make mistakes

Sperner's corridor to the same fixed point, on a gentle map and a twisted one. Two 32 by 32 labelled grids; the corridor crosses 44 triangles reading 100 corners on the gentle map and 148 triangles reading 204 corners on the twisted one.

What it costs to find the point that stays put

A random surface and the region above height 1. A smooth Gaussian random field on a 64 by 64 torus and its excursion set above 1: 14 pieces, 0 holes, Euler characteristic 14; expected 13.8.

Pieces minus holes on a random surface

A loop that cannot be undone, and the surface it bounds. The loop aba⁻¹b⁻¹ on the figure eight, which cannot be pulled tight, beside a square whose boundary reads the same word, showing that the loop bounds a surface.

What homology forgets about a loop

Pascal's triangle mod 2, and the room for the 2-dimensional projective space. Pascal's triangle with odd entries filled, rows 0 to 15, with row 3 marked as the tangent ledger 1 + a + a² and row 1 as the normal ledger 1 + a.

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