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--- blog:2023-05-02 [2023/05/02 22:32] – created pzhou
+++ blog:2023-05-02 [2023/06/25 15:53] (current) – external edit 127.0.0.1
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 Trefoil knot? Alexandre polynomial? Look at the Alexandre polynomial for trefoil, we discover that
 $$ \frac{x - x^{-1}}{\Delta_{3^1}(x^2)} = \sum_m \epsilon_m (x^m - x^{-m}) $$
+. There are 3-manifold invariants
+$$ F_K(x,q) = \hat Z(S^3 \RM K) \frac{ADO}{Alexander}$$
+expansion at $q=1$, gives Vasieliev invariant
+Quantum $A$-polymial!!
+$$ A(x,y) F_K(x,q) = 0. $$
+. Verma, R-matrix,  quantum group. Sunghyuk Park. https://arxiv.org/pdf/2004.02087.pdf
+. Gukov-Manolescu: https://arxiv.org/pdf/1904.06057.pdf
+. Lickorish-Wallace, any 3-manifold can be constructed from $S^3$ from a knot.
+It is a cool fact, that, by capping-off the knot. And there are Kirby move, invariant.
+===== Giovanni Felder: 2017 talk, on elliptic quantum group =====
+. shuffle product on symmetric functions. Three versions of theta functions
+$$ \theta(t) $$
+vanishes on the origin in the abelian group $\C, \C/\Z, \C/(\Z + \tau \Z)$.
+? What is the (odd) Jacobi Theta function? $\theta(z; \tau)$ is a function that is periodic in $z+1$, but quasi-periodic in $\tau$, like
+$$ \theta(z + \tau; \tau) = e^{-\pi i(\tau + 2 z)} \theta(z; \tau). $$