TITLE: multiple zeta values and knots invariants
QUESTION [7 upvotes]: 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?

REPLY [2 votes]: 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.