Matpic
mathematics, in pictures
Concept

Crossing number

The fewest pairs of edges that must cross in any drawing of a graph in the plane. It is zero exactly for planar graphs, and the first two graphs with crossing number one are the obstructions Kuratowski named.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

the trefoil. the trefoil, drawn as a closed curve with 3 crossings. At each crossing the strand passing underneath is broken, which is the only information the flat picture carries that the curve alone does not.

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.

topology · Knots Two graphs that will not lie flat, and one that will. K4, K5 and K3,3 in the best straight-line drawings a search could find. K4 has no crossings; the other two have one each, and Euler's formula shows that none can have none.

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.

discrete · Planarity the Hopf link, with every crossing signed. A diagram of the Hopf link with the under-strand broken at each crossing and each crossing between two components marked with its sign, which add to twice the linking number.

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.

topology · Linking number

Named alongside it

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

Topological invariantKnotOrientationReidemeister movesComplete graphCounting argumentDeformationEquivalenceEuler formulaGraphGraph colouringInvariant

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