# Knot at Lunch on October 23 2008

## Invitation

Dear Knot at Lunch People,

We will have our next lunch on Thursday October 23, at the (new and hopefully temporary) usual place, the seminar room on the 10th floor of 215 Huron St., at 12 noon.

As always, please bring brown-bag lunch and fresh ideas. I'm hoping to make a little map-handout, roughly showing the logical relations between all those things I am confused about. If the map will be ready by Thursday, we may talk about it. If not, we'll have to find something else to talk about.

## The Gospel in $v$, $w$ and Seven $Z$'s
 $Z:B\to{\mathcal A}^h$ Classifies braids! Detected by Lie algebras! 1 $Z:{\mathcal K}(\bigcirc)\to{\mathcal A}(\bigcirc)$ Related to metrized Lie algebras! This is the Kontsevich integral! Related to Chern-Simons theory and configuration space integrals! Has GPV formulas! 2 $Z:{\mathcal K}^{TG}\to{\mathcal A}^{TG}$ Enables "Algebraic Knot Theory"! Finitely presented! Related to Drinfel'd associators! No-go theorems in bounded degrees. Extremely challenging to compute. Leads to "internal quotients"! Still challenging to compute. $Z:pT\to{\mathcal A}^{pT}$ Does not exist. The modifier v means "virtual". Chords become arrows, planar algebras become circuit algebras and $4T$ becomes $6T$. "Algebraic Knot Theory" concepts still apply! The modifier w means "welded" or "weakly virtual". Overcrossings commute, tails commute. 3 $Z:wB\to{\mathcal A}^{wh}$ Ribbon tori in ${\mathbb R}^4$! Flying rings in ${\mathbb R}^3$! Easy formulas for $Z$! Relates to $\operatorname{Aut}(F_n)$! Classifies w-braids! 4 $Z:w{\mathcal K}(\bigcirc)\to{\mathcal A}^w(\bigcirc)$ Related to the Alexander polynomial! Related to general Lie algebras! Related to co-commutative Lie bi-algebras! Related to BF theory! 5 $Z:w{\mathcal T}\to{\mathcal A}^{wT}$ Still easy! Related (by other means) to free Lie algebras, derivations, traces! 6 $Z:w{\mathcal TT}\to{\mathcal A}^{wTT}$ Related to the Kashiwara-Vergne conjecture and the orbit method! Via Alekseev-Torrosian, explicit tree-level formulas for associators! 7 $Z:v{\mathcal TT}\to{\mathcal A}^{vTT}$ Related to general Lie bialgebras, Etingof-Kazhdan and quantum groups in general!