Combed Trisection Diagrams and Non-Semisimple 4-Manifold Invariants

Given a triple $H$ of (possibly non-semisimple) Hopf algebras equipped with pairings satisfying a set of properties, we describe a construction of an associated smooth, scalar invariant ${\tau }_H(X,\pi )$ of a simply connected, compact, oriented $4$-manifold $X$ and an open book $\pi$ on its boundary. This invariant generalizes an earlier semisimple version and is calculated using a trisection diagram $T$ for $X$ and a certain type of combing of the trisection surface. We explain a general calculation of this invariant for a family of exotic 4-manifolds with boundary called Stein nuclei, introduced by Yasui. After investigating many low-dimensional Hopf algebras up to dimension 11, we have not been able to find non-semisimple Hopf triples that satisfy the criteria for our invariant. Nonetheless, appropriate Hopf triples may exist outside the scope of our explorations.