Quantum invariants of closed framed $3$-manifolds based on ideal triangulations

We construct a new type of quantum invariant of closed framed $3$-manifolds with the vanishing first Betti number. The invariant is defined for any finite dimensional Hopf algebra, such as small quantum groups, and is based on ideal triangulations. We use the canonical element of the Heisenberg double, which satisfies a pentagon equation, and graphical representations of $3$-manifolds introduced by R. Benedetti and C. Petronio. The construction is simple and easy to be understood intuitively; the pentagon equation reflects the Pachner $(2,3)$ move of ideal triangulations and the non-involutiveness of the Hopf algebra reflects framings. For an involutory Hopf algebra, the invariant reduces to an invariant of closed combed $3$-manifolds. For an involutory unimodular counimodular Hopf algebra, the invariant reduces to the topological invariant of closed $3$-manifolds which is introduced in our previous paper. In this paper we formalize the construction using more generally a Hopf monoid in a symmetric pivotal category and use tensor networks for calculations.