Inverted Habiro series

Quantum algebra provides an important source of invariants of knots and 3-manifolds. Using Verma modules of the algebra $U_q(sl_2)$, Park defined a new quantum knot invariant (building on a previous work of Gukov–Manolescu) and observed that it can be written in a very peculiar form, which he called inverted Habiro series. I will describe a commutative ring $\Lambda$ that contains these series and explain how $\Lambda$ could arise as the center of some form of $U_q(sl_2)$. Then I will show how $q$-series identities of Euler, Hecke–Rogers and Ramanujan follow from the study of residues of the inverted Habiro series for the simplest knots. Finally, I will present some recent developments about the topological significance of these invariants.