Crossing number
Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.
Three moves, and what they cannot undo
A knot is a closed loop of string, and two knots are the same if one can be wiggled into the other. Reidemeister reduced all possible wiggling to three local pictures — which is what makes it possible to prove that a knot is knotted.
Two graphs that will not lie flat
Five points, every pair joined: no matter how the points are placed or how the lines are drawn, two of the lines cross. The proof is not about drawing at all — it counts edges against faces and finds one edge too many.
Two loops and one number
Give each crossing between two closed curves a sign, add them up, halve — and the answer does not depend on how the curves were drawn, how they are pushed about, or which way the picture was projected.
The crossings an alternating knot cannot lose
Peter Guthrie Tait drew knots for years and believed, without proof, that a diagram whose crossings alternate over and under, and which has no twist that can be undone, is already drawn with the fewest crossings the knot allows. The proof took a century, and when it came it needed only the two most extreme ways of smoothing the diagram and Euler's count of the regions of a map.
The crossings a graph cannot avoid
Five points all joined need one crossing, six need three, seven need nine. Euler's formula gives a lower bound that grows like the square of the number of points and is badly wrong; a sampling trick turns the same formula into a bound that grows like the fourth power and is right to within a constant. And for drawings with straight edges the count turns out to be something else entirely: the number of quadrilaterals the points make.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
KnotInvariantProjectionTopological invariantComplete graphEuler formulaKnot determinantOrientationPlanar graphReidemeister movesWritheAlternating knot