Concept

Deformation

A continuous movement of one object into another, with no cutting and no gluing along the way. What survives every deformation is what topology studies, and a quantity unchanged by all of them is called an invariant.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

3 loops in one ring, and the number that separates them. Loops drawn in an annulus, each labelled with how many times it goes round the hole. Loops with different counts cannot be deformed into one another without leaving the ring.

A loop that cannot be pulled tight

A hole is a strange thing to point at, because it is precisely where the surface is not. What can be pointed at is a loop of string lying on the surface — and the hole announces itself by refusing to let that loop be pulled in to a point.

topology · homotopy

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

Named alongside it

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

Topological invariant Winding number Continuity Covering space Crossing number Fundamental group Genus Homotopy Invariant Knot Linking number Loop

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