Matpic
mathematics, in pictures
Concept

Linking number

A whole number attached to two disjoint closed loops in space, counting with signs how often one passes through a surface the other bounds. No deformation keeping the loops apart can change it, so a non-zero value proves they cannot be separated.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

the Hopf link, with every crossing signed. A diagram of the Hopf link with the under-strand broken at each crossing and each crossing between two components marked with its sign, which add to twice the linking number.

Two 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.

topology · Linking number a loop that dips through and back, and the punctures of the disc. A link drawn with a shaded disc spanning the first loop, seen at an angle, with every place the second loop passes through the disc marked with the direction it was travelling in.

Zero can mean two different things

The linking number counts how often one loop pierces a surface the other one bounds. Two punctures of opposite sign add to nothing, and a loop that never goes through adds to nothing as well — so the answer zero is two pictures wearing one number.

topology · Linking number the Borromean rings, with every crossing signed. A diagram of the Borromean rings with the under-strand broken at each crossing and each crossing between two components marked with its sign, which add to twice the linking number.

Linked, 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.

topology · Linking number Alexander's horned sphere at stage 3. A tree of clasped pairs of horns, 7 of them, each pair's two circles passing once through the other's disc; the horns shrink geometrically and their tips converge.

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.

topology · Jordan curve 5 circles filling a three-sphere, every pair linked once. Several closed curves in space, nested on tori of different sizes, each pair passing through the other exactly once and none of them touching.

The circles that fill a three-sphere

A three-sphere is filled by circles — one through every point, no two meeting, every two linked exactly once. Stereographic projection is the only way anybody sees it, and the projected picture is a nest of circles on tori whose linking can be counted off the drawing.

topology · Stereographic projection

Named alongside it

The objects these essays reach for when they reach for this one.

KnotWinding numberOrientationBorromean ringsBrunnian linkCantor setCommutatorCounterexampleCrossing numberDeformationDimensionFibration

All concepts