Concept

Knot determinant

The absolute value of a knot's Alexander polynomial at −1, a whole number that every diagram of the knot gives alike. It counts colourings of the knot with a prime number of colours in the sense that a prime dividing it permits non-trivial ones.

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

How many colourings each knot allows. Three knots, and the number of ways their arcs can be coloured with three, five and seven colours under the crossing rule, beside the determinant computed separately from the same crossings.

Colours that count more than three

Three colours prove the trefoil is knotted and say nothing at all about the figure-eight, which refuses them exactly as an unknotted loop does. The repair is to stop colouring and start counting — with five colours, or seven, and with the arithmetic done modulo the number of them.

topology · Knots

The Alexander matrix of the trefoil. The trefoil with its 3 arcs numbered and its 3 crossings lettered, beside the 3 by 3 matrix they give. A minor of the matrix is the Alexander polynomial t − 1 + t⁻¹, whose value at −1 is the determinant 3.

A polynomial behind the colourings

The figure-eight knot and the cinquefoil both have determinant five, so they admit exactly the same colourings, and every counting argument treats them as one. Put a variable where the colouring rule has a two and the determinant becomes a polynomial — and the two knots come apart.

topology · Knots

Named alongside it

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

Invariant Knot Linear system Reidemeister moves Counting argument Covering space Decidability Determinant Modular arithmetic Polynomial Topological invariant Tricolourability

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