Bar-Natan Modules and Bar-Natan Pairings of Oriented 3-Manifolds

We discuss sesquilinear pairings defined by Bar-Natan modules (and their generalizations using general Frobenius algebras), which descend from universal manifold pairings recently discussed by Calegari, Freedman, Walker and others. Such a Bar-Natan pairing exists for each oriented closed surface with an embedded oriented closed 1-manifold (and each Frobenius algebra with involution). We also discuss how the Heegaard genus of closed 3-manifolds naturally appears in the calculation of Bar-Natan modules, and more generally how the calculation of Bar-Natan modules is related with the geometric topology of the 3-manifold.