Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Saturday, September 09, 2006

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 3x5 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.


Blogger Mahndisa S. Rigmaiden said...

09 10 06

Those damnable trefoils!

September 10, 2006 9:49 PM  
Blogger 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

September 12, 2006 10:39 PM  

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