### M Theory Lesson 177

Note that an intersection on the triangle plane arrangement becomes a square face on the cube. A (directed) cone from the top vertex will pick out the central horizontal edge of the cube, with the central point of the hexagon at one end representing the triangle. Observe that the number of edges in corresponding diagrams (planar arrangements to graphs) remains unchanged, whereas faces become vertices and vertices become faces. That is, this is a kind of Poincare duality.

## 2 Comments:

Kea, I've been having way too much fun with knots the last few days, particulary of the type where the cord weaves itself over and under, and where the knot can be tied with a single line.

These correspond to paths in the plane (or more generally, a 2-manifold, see this example of a mobius knot) that self intersect only once at any given point, as your figure shows, but with all the lines being the same line at infinity.

Hi Carl. Yes, I enjoyed your post. It reminded me of the old sailors' knot books I used to see in the quiet countryside libraries. From my brother and friends I know that sailors and knitters know more about knots than mountaineers, who only need to know a few simple handy ones.

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