Cerf Diagrams and Hatcher-Wagoner Invariants for Barbell Maps

For a half-unknotted implanted $(i,n-i)$-barbell $\beta=\beta_{i,n-i}$ in $M^n$, we construct two specific pseudo-isotopies, which we denote by standard barbell pseudo-isotopies, both resulting in that barbell diffeomorphism, each having a Cerf diagram only containing a single eye and with easily computable Hatcher-Wagoner invariants. We give an explicit formula for $\beta_{2,n-2}$ and a special class of $\beta_{3,n-3}$. Using this we show that for $n\geq 6$, every pseudo-isotopy with vanishing first Hatcher-Wagoner invariant can be isotoped to a composition of standard barbell pseudo-isotopies with $i=2$ or $3$. In dimension $n=4$, we further generalize the constructions and computations to half-unknotted immersed barbell diffeomorphisms and prove that for every $s\in \mathbb{Z}_2, \sigma\in \pi_2 M,\gamma\in \pi_1 M$ with $s=0 \text{ or }w_2^M(\sigma)\neq0$, there exists a standard immersed barbell pseudo-isotopy $f_\beta$ with the second induced Hatcher-Wagoner invariant $\Theta(f_\beta)=(s,\sigma)\cdot [\gamma]$.