# Knot at Lunch, May 24, 2007

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