Matpic
mathematics, in pictures
Ladder

Linking number — the ladder

3 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    rung 1 · topology
  2. 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.

    rung 2 · topology
  3. 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.

    rung 3 · topology

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