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

[math]\displaystyle{ 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) }[/math]

• Simplifying [math]\displaystyle{ e^{b+c}-e^be^c }[/math] using frozen feet, this becomes

[math]\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) }[/math]

• Alternatively,

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