Quantum invariants of framed 3-manifolds and the universal invariant of knots

For a finite-dimensional Hopf algebra, we construct an invariant of closed framed 3-manifolds with vanishing first Betti number via ordered ideal triangulations. For the small quantum group of the Borel subalgebra of $sl_2$, this invariant is expected to coincide with the $SO(3)$-WRT invariant multiplied by the order of the first homology group, up to a power of $q$. Although the construction has not yet been extended to 3-manifolds with boundary, it can be applied to knot complements, yielding a version of the universal invariant of knots. We will also briefly discuss possible directions toward extending this framework to the LMO invariant and the Kontsevich integral. Part of this work is based on joint work with S. Mihalache and Y. Terashima, and also with Y. Ota.