Concept

Alexander polynomial

A polynomial in one variable computed from a knot diagram's crossings, unchanged by every way of redrawing the knot. Its value at −1 is the determinant that the colourings count, and two values of it are enough to tell apart the small knots that random loops tie.

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

Seven points in space and a knotted path through all of them, seed 12. A projection of K₇ with straight edges, faint, and one Hamiltonian cycle drawn heavy with its crossings broken: a trefoil. 9 of the 360 Hamiltonian cycles are knotted.

Seven points and a knot they cannot avoid

Put seven points anywhere in space and join every pair with a straight segment. There are 360 closed paths that visit all seven points once each, and at least one of them is knotted — however the points are placed. The reason is a parity, as it was for six points and a linked pair: a number read off each path's knot adds up, over all 360, to something odd. Six points are not enough; seven always are.

topology · Linking number

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.

Knot Asymptotics Complete graph Connected sum Exhaustive search Hamiltonian cycle Intrinsic knotting Knot determinant Modular arithmetic Parity Random walk Stick number

All concepts