Concept

Simple closed curve

A loop drawn on a surface that never crosses or touches itself. On a torus the loops of this kind that cannot be shrunk away are exactly the classes (p, q) whose two counts share no common factor.

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

Loops on a torus that never cross themselves. Squares with opposite edges glued, each carrying one straight loop of a different slope, each labelled with its two crossing counts.

The loops on a torus that never cross themselves

Every loop on a torus is classified by two whole numbers: how often it goes round one way and how often the other. Some classes can be drawn without the loop ever crossing itself and some cannot, and the rule is the oldest in arithmetic — the two numbers must have no common factor. The same two numbers say how often any two loops must meet.

topology · Homotopy

A twist along the horizontal loop, done once. Squares with opposite edges glued and a shaded horizontal band, showing one loop before and after the torus is twisted along the band.

A twist that carries one loop to another

Cut a torus along a loop, turn one side of the cut once round, and glue it back. Nothing is torn, so every loop that did not cross itself still does not — but a loop of class (0, 1) is now a loop of class (1, 1). Two such twists reach every loop that never crosses itself, by Euclid's algorithm, and the symmetries they generate are exactly the whole-number matrices of determinant one.

topology · Homotopy

Named alongside it

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

Invariant Torus Coprime Dehn twist Euclidean algorithm Fundamental group Homeomorphism Homotopy Intersection number Lattice Mapping class group Matrix

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