Cerf Diagrams and Hatcher-Wagoner Invariants for Barbell Maps

For a half-unknotted implanted barbell $\beta$, we construct two specific pseudo-isotopies, both resulting in that barbell diffeomorphism, and compute the Hatcher-Wagoner invariants for both. We further generalize the results to half-unknotted immersed barbell diffeomorphisms and prove that for every $\sigma \in \pi_2 M$ and $\gamma \in \pi_1 M$, there is a half-unknotted immersed barbell diffeomorphism $\phi_\beta$ with the second induced Hatcher-Wagoner invariant $\Theta(\phi_\beta) = (0, \sigma) \cdot [\gamma]$.