On Handlebody Structures of Rational Balls

TL;DR

This paper proves that two previously defined families of rational balls bounding certain lens spaces are actually the same, using handle decompositions and Stein fillings to establish their equivalence.

Contribution

It demonstrates the equivalence of two different constructions of rational balls bounding lens spaces, resolving a question posed by Kadokami and Yamada.

Findings

The families of rational balls $B_{p,q}$ and $A_{m,n}$ are identical.

Each $A_{m,n}$ admits a Stein filling of the standard contact structure.

The result confirms the handlebody descriptions of these rational balls are consistent.

Abstract

It is known that for coprime integers $p>q\geq 1$, the lens space $L(p^2,pq-1)$ bounds a rational ball, $B_{p,q}$, arising as the 2-fold branched cover of a (smooth) slice disk in $B^4$ bounding the associated 2-bridge knot. Lekilli and Maydanskiy give handle decompositions for each $B_{p,q}$. Whereas, Yamada gives an alternative definition of rational balls, $A_{m,n}$, bounding $L(p^2,pq-1)$ by their handlebody decompositions alone. We show that these two families coincide - answering a question of Kadokami and Yamada. To that end, we show that each $A_{m,n}$ admits a Stein filling of the "standard" contact structure, $\bar{\xi}_{st}$, on $L(p^2,pq-1)$ investigated by Lisca.