Knot at Lunch, May 24, 2007: Difference between revisions
From Drorbn
Jump to navigationJump to search
No edit summary |
No edit summary |
||
| Line 9: | Line 9: | ||
* <math>[abw]=[baw]</math> if <math>|w|\geq 2</math> and similar identities. |
* <math>[abw]=[baw]</math> if <math>|w|\geq 2</math> and similar identities. |
||
* <math>\phi(a,b,c)=\phi(a,b)</math>, <math>\Phi=\exp\phi</math> and in our case, this is just <math>1+\phi</math>! |
* <math>\phi(a,b,c)=\phi(a,b)</math>, <math>\Phi=\exp\phi</math> and in our case, this is just <math>1+\phi</math>! |
||
* <math>[a^nb^mab]=(-1)^{n+m}[ab]a^nb^m</math>. |
|||
* <math>\phi=[ab]\lambda(a,b)</math>. With this, we have as follows: |
|||
* Unitarity becomes <math>\lambda(a,b)=\lambda(b,a)</math>. |
|||
* <math>\Phi^{312}=1+[ca]\lambda(ac)</math> where <math>c=-a-b</math>. |
|||
* Likewise for all other terms in the hexagon, which becomes |
|||
{{Equation*|<math>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>e^{b+c}-e^be^c</math> using frozen feet, this becomes |
|||
{{Equation*|<math>[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, |
|||
{{Equation*|<math>[ab]\left(\ldots\right) = [ab]\left(\lambda(a,b)e^{b+c}+\lambda(b,c)e^c+\lambda(a,c)\right)</math>}} |
|||
Latest revision as of 12:46, 24 May 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 [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
- Simplifying [math]\displaystyle{ e^{b+c}-e^be^c }[/math] using frozen feet, this becomes
- Alternatively,
