Maximal universal invariants from finite quotients of Verma modules

Abstract

We construct a sequence of new universal quantum knot invariants that are lifts of both the semi-simple and non semi-simple $U_q(sl_2)$ quantum knot invariants. More specifically, for any level $\mathscr N$ we define a ``level $\mathscr N$ universal invariant'' $ \widetilde{\Omega}_{\mathscr N}(L)$ arising from quantum traces on finite quotients of the generic Verma module over certain quotient rings. We show that for $\mathscr N$ prime, this is the maximal invariant that can arise from the $\mathscr N$-part of the Verma module, and it is a specific interpolation between the $\mathscr N^{th}$ coloured Jones and $\mathscr N^{th}$ ADO polynomials. For $\mathscr N$ non prime $ \widetilde{\Omega}_{\mathscr N}(L)(q,s)$ has a richer structure, it recovers the $\mathscr N^{th}$ coloured Jones and $\mathscr N^{th}$ ADO polynomials, but it could contain more information which is not seen in the sequence of coloured Jones and ADO invariants.