Knot at Lunch, May 24, 2007

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First meeting for summer 2007! Peter Lee is telling us about associators with frozen feet. See also his handout from the CMS Winter 2006 Session on Knot Homologies - front: , back: and Dror's very partial paperlet, Associators with Frozen Feet.

• Definition of ${\mathcal A}^n$ and its relation with finite type invariants.
• The frozen feet quotient.
• The action of $\Delta$, $\eta_i$, and of $\star\mapsto\star^{23}$, etc. (Dror: see also VS, TS and TG Algebras.)
• Generators and relations: $R^\pm$, $\Phi^\pm$, the hexagons and pentagon, unitarity, non-degeneracy, group-like property.
• $[abw]=[baw]$ if $|w|\geq 2$ and similar identities.
• $\phi(a,b,c)=\phi(a,b)$, $\Phi=\exp\phi$ and in our case, this is just $1+\phi$!
• $[a^nb^mab]=(-1)^{n+m}[ab]a^nb^m$.
• $\phi=[ab]\lambda(a,b)$. With this, we have as follows:
• Unitarity becomes $\lambda(a,b)=\lambda(b,a)$.
• $\Phi^{312}=1+[ca]\lambda(ac)$ where $c=-a-b$.
• Likewise for all other terms in the hexagon, which becomes

$e^{b+c}=(1+[ab]\lambda(a,b))e^b\cdots = e^be^c+[ab]\left(\lambda(a,b)e^{b+c}+\lambda(b,c)e^c+\lambda(a,c)\right)$

• Simplifying $e^{b+c}-e^be^c$ using frozen feet, this becomes

$[cb]\left(\frac{e^{b+c}-1-b-c}{b(b+c)}-\frac{e^c-1-c}{c}\right) = [ab]\left(\lambda(a,b)e^{b+c}+\lambda(b,c)e^c+\lambda(a,c)\right)$

• Alternatively,

$[ab]\left(\ldots\right) = [ab]\left(\lambda(a,b)e^{b+c}+\lambda(b,c)e^c+\lambda(a,c)\right)$