A Strange Weight System by Duzhin
May-18-11
9:04 AM
Date: Tue, 17 May 2011 10:25:52 +0400
From: Sergei Duzhin <...>
To: "D. Bar-Natan" <drorbn@math.toronto.edu>
Subject: kleinian weight systems
Dror,
About 10 years ago I told you the construction of a weight system for 1-3-valent graphs starting from a skew-symmetric function of 3 variables F(x,y,z) that satisfies the identity
F(x,y,z)*F(u,v,z)-F(x,u,z)*F(y,v,z)+F(x,v,z)*F(y,u,z)=0
(a typical example is det([f(x),f(y),f(z)],[g(x),g(y),g(z)],[h(x),h(y),h(z)]) , but there are others). To evaluate this weight system on a given diagram, you assign variables to edges, take the product over vertices of the values F(e1,e2,e3) for the three edges meeting at a vertex, and take the average over permutations (of inner and outer vertices separately).
You explained me that the resulting weight system is always equivalent to one coming from sl_2. Now I tried to reconstruct this argument, and I could not:( May I ask you to give a hint, if you still remember that?
Sergei