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mathematics, in pictures
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Things that cannot be done — page 8

Results that close a door rather than open one — and the peculiar difficulty of drawing a picture of something that does not exist.
Signature against unknotting, for every knot to seven crossings. A table of the fourteen prime knots with up to seven crossings, listing determinant, signature, the lower bound on crossing changes it gives, the number of changes found by search, and whether the two agree. Topology

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.

 Several colours on every vertex of a Kneser graph. A grid of Kneser graphs on pairs from five to eight points against the number of colours per vertex, each cell giving the fewest colours needed and the counting lower bound. Topology

Several colours on every vertex

Give every pair from six points three colours, so that pairs with nothing in common share no colour. Counting says nine colours might do; ten are needed. Stahl conjectured in 1976 exactly how many colours every such problem needs — a formula that meets Lovász's topological answer at one colour a vertex and the obvious answer at k — and a search over stars and triangles confirms it in every case small enough to run.

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