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: CMS 2006 Lee Handout Front.png, back: CMS 2006 Lee Handout Back.png 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)