Kirby diagrams for an infinite family of exotic $\mathbb{R}^4$'s

Abstract

Eli, Hom, and Lidman showed that the manifolds produced by attaching the simplest positive Casson handle $CH^+$ to a slice disc complement of the ribbon knot $T_{2,n}\#T_{2,-n}$ for $n\ge3$ and odd, and removing the boundary, form a countably infinite family of exotic $\mathbb{R}^4$'s. They provided a Kirby diagram for the case $n=3$. In this short note, we extend this for $n\ge3$ and odd, and provide Kirby diagrams for two such families of exotic $\mathbb{R}^4$'s, which are then shown to be equivalent. We then generalise these diagrams to a family of exotic $\mathbb{R}^4$'s built using ribbon disc complements of the pretzel knots $P(n,-n,2k)$.