multiple zeta values and knots invariants

Question

I have heard several times that MZV appear in the context of knot invariants and deformation quantisation. Could anyone explain how and give some references?

Answer 1

The knot invariants you're thinking of, like the Jones polynomial, come from braided monoidal categories deforming the representations of Lie groups. This sort of deformation, in general, was shown to exist by Drinfeld in the (rather dense) "On quasitriangular Quasi-Hopf algebras and a group closely connected with $Gal(\overline{Q}/Q)$" and the algebraic aspect is maybe at least a little clearer in Dror Bar Natan's "On Associators and the Grothendieck-Teichmuller Group I".

The analytic aspect comes from the monodromy of the KZ equations, which solves the associator problem and also allows for directly finding these invariants of a braid closure as monodromy of a differential equation along the braid; "Hyperplane Arrangements And Holonomy Equations" by De Concini and Procesi demonstrate this in the last section for the universal Vassiliev invariant.

These monodromies can be described in terms of the asymptotics of solutions of the equation $$\frac{d\psi}{dz}=\left(\frac{A}{z}+\frac{B}{z-1} \right)\psi $$ with $\psi$ being valued in the completion of the tensor algebra generated by $A$ and $B$; given the unique solutions (on a simply connected subset of $\mathbb{C} \setminus \{0,1\}$ containing the interval $(0,1)$) with asymptotics $\psi_0 \sim z^A$ and $\psi_1 \sim (1-z)^B$, the ratio $$\Phi(A,B)=\psi_1^{-1}\psi_0 $$ is a fixed element of the completed tensor algebra; if you try to compute this directly as an ordered integral you get integrals of the form $$\int_{0 < z_1 < z_2 < ... < z_n < 1} \omega_{z_1}\omega_{z_2}...\omega_{z_n} $$ where $\omega_{z_i}$ is either $dz_i/z_i$ or $dz_i/(1-z_i)$; expanding the terms of the latter form as $\sum_{j=0}^\infty z_i^j$ should give you the multiple zeta values, but I don't know a good reference for this last part.

Answer 2

If you told me that knot invariants, modular forms, zeta values and quantum field theory were all related, I would not argue or be the slightest bit surprised. And yet, these connections were only disovered in the early 1990s

• Values of Zeta Functions and Applications Don Zagier

At the very end, mentions a then-recent discover of Kontsevich on Vassiliev invariants and knot theory. And the Knizhnik-Zamolodichikov eq.

Stating these objects are related and pointing to a precise relationship are two different things. It is a topic currently under development.