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 2 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

Named alongside it

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

Knot Orientation Winding number Crossing number Deformation Homology Invariant Isotopy Link Loop Reidemeister moves Spanning surface

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