Turning number
Named by 2 essays across one field — each of them below, with the objects they name alongside it.
The turns a tangent makes
Drive once round a closed track and count how many full turns the steering has made. A circle is one turn, a figure-eight is nought, and every loop adds one. Whitney and Graustein proved that this one number decides when one track can be bent into another without ever stopping — and it can be read off the crossings of a drawing without following the curve at all.
How many ways a curve can cross itself
A closed curve that crosses itself three times can be drawn in exactly six essentially different ways on a sphere, and twenty in the plane. Four crossings give nineteen and eighty-two; six give 376 and 2,645. Every one of them is a word — the order in which the crossings are met — and a choice of direction at each crossing, and the whole census comes from asking which of those choices close up on a sphere.
Named alongside it
The objects these essays reach for when they reach for this one.
ImmersionBurnside lemmaEuler characteristicExhaustive searchFundamental groupHomeomorphismHomotopyParityPlanar graphTangentWinding number