Trippy Trefoils

Loday found a realisation of all the Stasheff polytopes, including the K4 one shown here. The vertices in 4 dimensions are permutations on 4 letters, so the polytope lies in the 3 dimensional plane x + y + z + w = 10 and the vertices are on the integer lattice.
Loday also noticed that by tracing a curve around the faces, once through the pentagons and twice through the squares, one obtains a trefoil knot by judiciously choosing the crossings at the centre of the squares.

The classical Alexander knot invariant for the trefoil is obtained as follows. Let (t,-t,1,-1) be labels for the 4 region types: such as before undercrossing on left etc. Make a 3×5 matrix for the 3 crossings and the 5 regions of the trefoil knot, such as

-t t -1 0 1
-1 t 0 -t 1
0 1 -1 -t t

Now ignore 2 columns, take the determinant, and scale by powers of t appropriately, to obtain A(trefoil) = (t^-1) – 1 + t. The Jones polynomial can distinguish handedness for the trefoil knot. The left Jones polynomial is J(trefoil) = t + t^3 – t^4. Recall that the trefoil knot appeared recently in a heuristic discussion of the Koide formula for mass ratios. Here one uses the variable t=exp(a), which is the usual change of variables for Vassiliev invariants.


2 Responses so far »

  1. 1

    Mahndisa S. Rigmaiden said,

    09 10 06

    Those damnable trefoils!

  2. 2

    QUASAR9 said,

    Hi kea, If Geometry is your thing
    you’ll enjoy
    dialogues with Eide
    (Geometry & Physics)

    His posts are simply great, he’ll welcome your comments, he’ll keep an eye on your posts, and he’ll be happy to take you to the next level

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