Concept

Connected sum

The knot made by cutting two knots open and joining their loose ends, so that one is tied after the other on a single loop. Genus and the determinant behave simply under it, and long random loops are mostly connected sums of small knots.

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

The gluing abab makes a projective plane. A polygon whose edges carry the word abab, with arrows for the direction each edge is glued and the corners coloured by which vertex they become.

A disc sewn to a Möbius band

The band has one boundary curve, and a disc has one boundary curve. Sew them together and the result is the smallest closed surface with one side — the one every other one-sided surface is built out of.

topology · Orientability

Three random loops and the knots they tie. 0₁: 12 crossings; 3₁: 18 crossings; 4₁: 16 crossings.

Which knots a random loop ties

Long random loops are almost always knotted, but knotted as what? Sorting 14,000 closed random polygons by two values of the Alexander polynomial answers it. The trefoil leads at every length, five times as common as the figure-eight at 400 sticks. Each knot's share, divided by the unknotted share, grows like a power of the length, and the trefoil tied twice grows about twice as fast as the trefoil — what a loop carrying independent small tangles would do. Up to 300 sticks the double trefoil turns up exactly as often as two independent trefoils should.

topology · Knots

Named alongside it

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

Alexander polynomial Antipodal map Asymptotics Classification Closed surface Cross-cap Euler characteristic Exhaustive search Gluing diagram Immersion Knot Knot determinant

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