On the Ozsváth-Szabó $d$-invariants for almost simple linear graphs

TL;DR

This work refines the Karakurt-Şavk computation of the Ozsváth-Szabó $d$-invariant for Brieskorn spheres $\Sigma(p,q,r)$ with almost simple linear graphs, providing an explicit max-formula for odd $p$ via a quadratic form $F_{p,q}$ and a refined index $D(p,q,r)$. It establishes parity- and growth-dependent criteria for when $d(\Sigma(p,q,r))$ equals $D(p,q,r)$ and constructs infinite families where equality holds or fails, yielding new infinite summands in the 3-dimensional homology cobordism group $\Theta_{\mathbb{Z}}^3$ and informing knot concordance through $\mathcal{C}_{TS}$. The paper also derives a parity-independent inequality guiding $D(p,q,r)$ across families, analyzes the Fibonacci and other infinite classes, and demonstrates concrete implications for braids, pretzel knots, and Dehn surgery realizations. Overall, it deepens the link between 3-manifold invariants and knot concordance, providing tools to generate and distinguish infinite families of non-cobordant Brieskorn spheres and their associated concordance classes.

Abstract

Karakurt and Şavk calculate the Ozsváth-Szabó $d$-invariant for Brieskorn homology $3$-spheres where surgery diagrams can be expressed as almost simple linear graphs. In this paper, we introduce a refiniment of Karakurt and Şavk's formula for these $d$-invariants. We also introduce infinite examples of inequalities that appear in this refinement in which the equality holds, and infinite examples in which it does not hold. We also discuss an application to knot concordance group.

Figures (6)

• Figure 1: A surgery diagram of $\Sigma(p,q,r)$ with $pq+pr-qr=1$.
• Figure 2: The blue area is the range where $d(\Sigma(p,q,r))$ could be calculated using KS20, and the red area is the range where $d(\Sigma(p,q,r))$ could be calculated using Proposition \ref{['thm:1-19']}. The green area is the range that there are eamples that satisfy $d(\Sigma(p,q,r)) \neq D(p,q,r)$. The green and pink areas are currently unexplored areas, except for the case where $\alpha_{p,q}=0$ (Theorem \ref{['thm:alpha0case']}). This figure shows that in addition to Corollary \ref{['cor:Fibonacci']}, this study has newly obtained pairs of Brieskorn homology $3$-spheres that are not homologically cobordant.
• Figure 3: Slam-dunk.
• Figure 4: A surgery diagram of $\Sigma(p,q,r)$. The right side of the diagram corresponds to a surgery diagram for a $3$-manifold obtained by performing resolution of $\Sigma(p,q,r)$, which resolves singularities in $\Sigma(p,q,r)$ separately from the method using Milnor fiber.
• Figure 5: A surgery diagram of $\Sigma(p,q,r)$ with $pq+pr-qr=1$.

Theorems & Definitions (119)

• Proposition 1.1: KS20
• Theorem : Theorem \ref{['thm:oddcase']}
• Remark 1.2
• Theorem : Theorem \ref{['thm:alpha0case']}
• Theorem : Theorem \ref{['thm:1-19']}
• Proposition : Proposition \ref{['prop: piqiri']}
• Theorem : Theorem \ref{['thm: su']}
• Theorem : Theorem \ref{['thm:irregularcondition']}
• Proposition : Proposition \ref{['prop: Fibonacci case']}
• Definition 2.1: Brieskorn homology 3-sphere