On chain link surgeries bounding rational homology balls and $χ$-slice 3-braid closures

TL;DR

This work classifies when surgeries on twisted chain links and closures of 3-braids yield rational homology 3-spheres that bound rational homology balls, via the cubiquity obstruction and Heegaard Floer d-invariants. It translates 3-braid closures into double branched covers of chain-link surgeries, and develops a lattice-theoretic framework with dual strings and explicit sets S_i to organize the analysis. The authors prove a nearly complete χ-slice–ribbon classification for QA 3-braid closures, identifying the exceptional Brieskorn family and establishing sharpness criteria for the associated 4-manifolds; they also provide a practical algorithmic method (via a SageMath notebook) to compute cubiquity. Overall, the paper advances understanding of which Q S^3 bound Q B^4 and extends Lisca’s slice–ribbon results from 3-braid knots to QA 3-braid links, with implications for the structure of double branched covers and L-spaces.

Abstract

We determine which integral surgeries on a large class of circular chain links bound rational homology balls. Our key tool is the lattice-theoretic cubiquity obstruction recently developed by Greene and Owens. We discuss a practical method of computing it, and, as an application, prove that a generalisation of the slice--ribbon conjecture holds for all but one infinite family of quasi-alternating 3-braid links. This extends previous results of Lisca concerning the conjecture for 3-braid knots.

Figures (20)

• Figure 1: Integral surgery along an $n$-component $t$-half twisted chain link $L_n^t$, which we denote by $S^3_{\textbf{x}}(L_n^t)$, where $\textbf{x}=(x_1,\ldots,x_n)$ and $x_i\in\mathbb{Z}$ for all $i$. The box labeled $t$ indicates the number of half-twists.
• Figure 2: Links whose double branched covers yield $Y^t_{\textbf{x}}$, $|t|\le1$.
• Figure 3: A $\mathbb{Q} B^4$ bounded by $S^3_{(a,0)}(L_2^t)$.
• Figure 4: The 4-manifold $X^t_{(a_1, \dots, a_n)}$ whose boundary is $Y^t_{(a_1, \dots, a_n)}$, the double cover of $S^3$ branched over the closure of the 3-braid $(\sigma_1\sigma_2)^{3t}\sigma_1\sigma_2^{-(a_1-2)}\dots\sigma_1\sigma_2^{-(a_n-2)}$.
• Figure 5: $\partial X=\partial Z$.

Theorems & Definitions (50)

• Proposition 1.1
• Proposition 1.2
• Remark 1.3
• Theorem 1.4: Theorem 1.7 in simone2020classification
• Theorem 1.5
• Theorem 1.6: murasugi
• Definition 1.7
• Proposition 1.8
• Remark 1.9
• Theorem 1.10