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This talk with start with a (re)introduction to a beautiful and unexpected application of the representation theory of quantum groups - that is to the creation of quantum knot invariants. The construction starts with a link whose strands have been colored by representations and produces a polynomial. Even for a knot as familiar as the trefoil, different colorings can produce dramatically different polynomials.

Remarkably, in some cases one can guarantee the existence of recursion relations in this color dependence by reframing the problem in terms of modules of the quantum Weyl algebra.

We will describe this algebraic approach and (time permitting) how the speaker is using it to study the ADO invariant.

This talk is based on current work with Tudor Dimofte and Nathan Geer.