Knot determinant
Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
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.
How many changes undo a knot
Cut the string at a crossing, pass it through the other strand and join it up again, and any knot can be undone by doing that often enough. The fewest changes needed is the unknotting number, and proving that fewer will not do needs a number that each change can move only a little. The signature moves by at most two per change — enough to settle thirteen of the fourteen knots up to seven crossings, and not the fourteenth.
Six sticks tie a trefoil, and five cannot
Build a knot from straight sticks joined end to end and ask for the fewest. A trefoil takes six, and the six corners can be whole-number points in a box ten units wide. Five sticks can cross one another five times in a picture, as often as a cinquefoil needs, and still tie nothing — the five crossings always twist three one way and two the other. The fewest sticks is a measure of how knotted a knot is that no diagram shows directly.
Almost every long loop is knotted
Close a random walk into a loop and ask whether it is knotted. With ten steps almost never; with a hundred, more than one time in five it can be proved knotted by a single number; with two hundred and fifty, more than half. The chance of staying unknotted falls exponentially with length — Frisch, Wasserman and Delbrück guessed it for polymer rings around 1961, and it was proved in 1988 — because a knot needs only one small tangle somewhere, and a long loop has room for many.
Three sines tie a knot
Let a point move with a different sine wave in each of the three directions of space, the three frequencies whole numbers with no common factor, and it traces a closed curve that is usually knotted. With frequencies 2, 3 and 7 the curve is the knot 5₂. The trefoil, the simplest knot of all, can never be made this way: the half-turn that shifting time by half a period performs forces a condition on the knot's polynomial that the trefoil fails.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
KnotInvariantModular arithmeticCrossing numberLinear systemProjectionRandom walkReidemeister movesAlexander polynomialAsymptoticsChiralityConnected sum