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 $1+\phi$!
• $[a^{n}b^{m}ab]=(-1)^{n+m}[ab]a^{n}b^{m}$.
• $\phi =[ab]\lambda (a,b)$. 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

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

• Simplifying $e^{b+c}-e^{b}e^{c}$ 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,

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