Knot at Lunch, May 24, 2007: Difference between revisions
From Drorbn
Jump to navigationJump to search
No edit summary |
No edit summary |
||
| (2 intermediate revisions by the same user not shown) | |||
| Line 5: | Line 5: | ||
* Definition of <math>{\mathcal A}^n</math> and its relation with finite type invariants. |
* Definition of <math>{\mathcal A}^n</math> and its relation with finite type invariants. |
||
* The frozen feet quotient. |
* The frozen feet quotient. |
||
* The action of <math>\Delta</math> and of <math>\star\mapsto\star^{23}</math>, etc. ({{Dror}}: |
* The action of <math>\Delta</math>, <math>\eta_i</math>, and of <math>\star\mapsto\star^{23}</math>, etc. ({{Dror}}: see also [[VS, TS and TG Algebras]].) |
||
* Generators and relations: <math>R^\pm</math>, <math>\Phi^\pm</math>, the hexagons and pentagon. |
* Generators and relations: <math>R^\pm</math>, <math>\Phi^\pm</math>, the hexagons and pentagon, unitarity, non-degeneracy, group-like property. |
||
* <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>[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 and its relation with finite type invariants.
- The frozen feet quotient.
- The action of , , and of , etc. (Dror: see also VS, TS and TG Algebras.)
- Generators and relations: , , the hexagons and pentagon, unitarity, non-degeneracy, group-like property.
- if and similar identities.
- , and in our case, this is just !
- .
- . With this, we have as follows:
- Unitarity becomes .
- where .
- Likewise for all other terms in the hexagon, which becomes
![{\displaystyle [abw]=[baw]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f51aeb2fc7573f61bae143f67d016aff3d3a0f46)
![{\displaystyle [a^{n}b^{m}ab]=(-1)^{n+m}[ab]a^{n}b^{m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f47fcf0b4ccac631df9d6358273cbd51b79664fe)
![{\displaystyle \phi =[ab]\lambda (a,b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e21c60abb966f33a02cfc173be130c6787a918)
![{\displaystyle \Phi ^{312}=1+[ca]\lambda (ac)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f3576f87fd8c3de8d4b0e360bd0942a55d21b72)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff2dd698b3867339473329dd7de676505fb5c1ac)
- Simplifying 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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9182ba5a319310f4db765bd34bc76c36449ba198)
- Alternatively,
![{\displaystyle [ab]\left(\ldots \right)=[ab]\left(\lambda (a,b)e^{b+c}+\lambda (b,c)e^{c}+\lambda (a,c)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/edae64378e28c35370ada45de4583166702dce43)
