Matpic
mathematics, in pictures
Series

Knots — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · topology
  2. 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.

    part 2 · topology
  3. 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.

    part 3 · topology
  4. The Seifert circles of the trefoil. The trefoil with an orientation, cut at each of its 3 crossings and reconnected the way the orientation allows. The 6 segments form 2 circles, and the surface built from them has genus 1.

    The surface a knot bounds

    Every knot is the edge of a surface with two sides, and Seifert found a way to build one from any diagram: smooth the crossings, fill the circles that result with discs, and join them with twisted bands. Counting the handles gives an upper bound on how complicated the knot is, the Alexander polynomial gives a lower one, and for the simplest knots the two meet.

    part 4 · topology

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