Cerf Diagrams and Hatcher-Wagoner Invariants for Barbell Maps

For a half-unknotted implanted barbell $β$, 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 $σ\in π_2 M,γ\in π_1 M$, there is a half-unknotted immersed barbell diffeomorphism $φ_β$ with the second induced Hatcher-Wagoner invariant $Θ(φ_β)=(0,σ)\cdot [γ]$.