The primary polytope for a typical hexagon secondary polytope is a wedge shape, or truncated tetrahedron. This system has a $3 \times 3$ Kasteleyn matrix. Primary polytopes in three dimensions correspond to $2 \times m$ arrays (ie. a set of $m$ $2$-vectors) in the secondary space. The wedge can be expressed as the array

$\{ (1,0),(1,1),(0,1),(-1,0),(-1,-1),(0,-1) \}$

which can be used to define an oriented string version of Grothendieck's dessins, using two rows for the standard two colours of the nodes and letting the $(0, \pm 1)$ entries define string orientation and winding numbers for six loops on a torus.

Recall that truncated tetrahedra also appear in Hodges' polytopes for twistor scattering theory. A wedge describes the first non trivial amplitude (of a certain type) and higher $n + k$ point amplitudes are described by gluing a double polygon (along a square) to the wedge to form a new tope.

The new Mason and Hodges KITP talks are now available online.

8 years ago

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