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

TL;DR

This paper extends Kirby diagram constructions for an infinite family of exotic 's, demonstrating their equivalence and generalizing to new families using ribbon disc complements of pretzel knots.

Contribution

It provides explicit Kirby diagrams for a broader class of exotic 's and proves their equivalence, expanding understanding of these manifolds.

Findings

Kirby diagrams for 's with n  and odd are extended.

Two families of exotic 's are shown to be equivalent.

Generalization to families using ribbon disc complements of pretzel knots P(n,-n,2k).

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 $P(n,-n,2k)$.