The Heegaard Floer d-invariant for more rational homology spheres

Isabella Khan

arxiv: 2605.05332 · v1 · submitted 2026-05-06 · 🧮 math.GT

The Heegaard Floer d-invariant for more rational homology spheres

Isabella Khan This is my paper

Pith reviewed 2026-05-08 15:34 UTC · model grok-4.3

classification 🧮 math.GT

keywords Heegaard Floer homologyd-invariantlattice homologyplumbed manifoldsrational homology spheresNémethi conjecturethree-manifold invariants

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The pith

Némethi's lattice homology formula gives the Heegaard Floer d-invariant for every negative-definite plumbed rational homology sphere.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that a combinatorial formula based on lattice homology computes the d-invariant, defined as the lowest absolute grading of a generator in HF^+(Y,s), for the entire class of negative-definite plumbed rational homology spheres. The argument invokes an existing isomorphism to transfer the computation from lattice homology directly to Heegaard Floer homology. This matters because it removes the previous restriction to special subclasses and supplies an explicit way to calculate the invariant from the plumbing graph data alone. A reader would care since the d-invariant encodes information about the manifold's smooth structure and its four-dimensional fillings.

Core claim

By applying Zemke's isomorphism between lattice homology and Heegaard Floer homology, the paper shows that Némethi's formula, which extracts the minimal grading from the lattice associated to the negative-definite plumbing, equals the d-invariant of the corresponding rational homology sphere for every spin^c structure.

What carries the argument

Zemke's isomorphism between lattice homology and Heegaard Floer homology, which identifies the minimal grading generator in one theory with the corresponding generator in the other.

If this is right

The d-invariant becomes computable by a finite algorithm that inspects only the weights and graph of the plumbing diagram.
All previously known special cases of Némethi's formula are recovered as instances of the general result.
The correction term in Heegaard Floer homology is now determined for the full family of negative-definite plumbed manifolds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same identification might permit lattice-based formulas for d-invariants of manifolds obtained by other surgeries if analogous isomorphisms hold.
Comparisons with Seifert fibered spaces whose d-invariants are already known by other methods could provide independent checks.
The result suggests lattice homology may capture additional Heegaard Floer data beyond the d-invariant in these cases.

Load-bearing premise

Zemke's isomorphism between lattice homology and Heegaard Floer homology applies without additional restrictions to every negative-definite plumbed rational homology sphere.

What would settle it

A direct computation of HF^+ for any specific negative-definite plumbed rational homology sphere whose d-invariant differs from the minimal grading given by Némethi's lattice formula.

read the original abstract

The Heegaard Floer d-invariant for a rational homology sphere Y and spin$^c$-structure $\mathfrak{s}$ is defined as the minimal absolute grading of a generator of $HF^+(Y; \mathfrak{s})$. In 2005, N\'emethi used lattice homology to compute the d-invariant for a particular class of negative-definite plumbed rational homology spheres, and conjectured that his formula should hold for all negative-definite plumbed rational homology spheres. In this paper, we use Zemke's isomorphism between lattice and Heegaard Floer homology to prove N\'emethi's conjecture.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Khan proves Némethi's conjecture for the d-invariant on all negative-definite plumbed rational homology spheres by applying Zemke's isomorphism.

read the letter

The main takeaway is that this short paper proves Némethi's conjecture from 2005 by using Zemke's isomorphism to show that the lattice homology formula gives the Heegaard Floer d-invariant for every negative-definite plumbed rational homology sphere. It is new in closing that gap for the complete class. Earlier work handled only particular plumbings, and this one uses the isomorphism to extend it without extra work. The paper does well by staying direct: it invokes the existing result and transfers the formula, with no circularity or invented terms. The only real question is whether Zemke's isomorphism applies to all the relevant plumbings. If it requires the graph to be a tree or imposes other conditions beyond negative-definiteness, then some cases might fall outside and the conjecture would not be fully proved. The abstract treats it as a direct proof, so the manuscript likely confirms the conditions are met for the general case. If that check is there and the gradings match properly, the argument is solid. This is for specialists in 3-manifold topology who care about explicit d-invariant formulas or lattice homology. It is not a big reorganization of the field but a useful completion of prior work. The math looks grounded in published theorems rather than new claims. I would bring it to a reading group focused on Floer homology to discuss the details of the isomorphism application. I would cite it in my own work if I needed the formula for these manifolds. It deserves peer review because the claim is clear and the evidence is the cited isomorphism. Recommendation: send it out for referees to verify the coverage and any grading transfers.

Referee Report

2 major / 2 minor

Summary. The manuscript proves Némethi's 2005 conjecture that the d-invariant of any negative-definite plumbed rational homology sphere Y with spin^c-structure s equals the value given by Némethi's lattice-homology formula. The argument invokes Zemke's isomorphism between lattice homology and Heegaard Floer homology to transfer the minimal grading computation from the lattice side to HF^+(Y;s).

Significance. If the isomorphism applies without further restrictions, the result resolves the conjecture for the entire class of negative-definite plumbed rational homology spheres and supplies a uniform computational method for their d-invariants. The proof strategy is economical, relying on an existing isomorphism rather than new technical machinery.

major comments (2)

[main argument / proof of the conjecture] The central claim rests on Zemke's isomorphism holding for every negative-definite plumbing graph that yields a rational homology sphere. The manuscript must cite the precise statement of Zemke's theorem (including any hypotheses on the graph or intersection form) and verify that all such plumbings satisfy those hypotheses; if extra conditions such as the graph being a tree are required, the proof would leave part of the conjecture unproven.
[application of the isomorphism] The transfer of generators and absolute gradings under the isomorphism must be checked explicitly for the minimal-degree generator that defines the d-invariant; any mismatch in grading conventions or support of the isomorphism would invalidate the equality with Némethi's formula.

minor comments (2)

Add equation numbers to the displayed statement of Némethi's lattice formula and to the definition of the d-invariant for easy cross-reference.
Clarify in the introduction whether the result applies only to integral homology spheres or to all rational homology spheres obtained by negative-definite plumbings.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the significance of our result. We address each major comment below and will incorporate the requested clarifications in a revised manuscript.

read point-by-point responses

Referee: The central claim rests on Zemke's isomorphism holding for every negative-definite plumbing graph that yields a rational homology sphere. The manuscript must cite the precise statement of Zemke's theorem (including any hypotheses on the graph or intersection form) and verify that all such plumbings satisfy those hypotheses; if extra conditions such as the graph being a tree are required, the proof would leave part of the conjecture unproven.

Authors: We agree that the precise statement of Zemke's theorem must be cited explicitly, including its hypotheses. In the revised manuscript we will quote the relevant theorem from Zemke's work verbatim, noting that it applies to all negative-definite plumbings yielding rational homology spheres (no tree condition is required). We will add a short verification paragraph confirming that every such plumbing graph satisfies the stated hypotheses on the intersection form and the resulting manifold, thereby covering the full class in Némethi's conjecture. revision: yes
Referee: The transfer of generators and absolute gradings under the isomorphism must be checked explicitly for the minimal-degree generator that defines the d-invariant; any mismatch in grading conventions or support of the isomorphism would invalidate the equality with Némethi's formula.

Authors: We will expand the argument in the revised version to include an explicit check of how Zemke's isomorphism maps the minimal-grading generator of the lattice homology to the corresponding generator in HF^+. This discussion will address the absolute grading conventions and confirm that the support of the isomorphism preserves the minimal degree, ensuring the d-invariant equals the value given by Némethi's lattice-homology formula. revision: yes

Circularity Check

0 steps flagged

No circularity; proof applies external isomorphism to transfer known formula

full rationale

The manuscript's central step invokes Zemke's independently published isomorphism to equate lattice homology with Heegaard Floer homology, thereby extending Némethi's lattice-based formula for the d-invariant to all negative-definite plumbed rational homology spheres. This citation is to prior work by a different author and is treated as an external theorem rather than derived or fitted inside the paper. No equations reduce the claimed result to a self-definition, a fitted parameter renamed as prediction, or a self-citation chain; the derivation chain remains open to the external benchmark and does not collapse by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of Zemke's isomorphism to the manifolds in question together with the standard definitions of the d-invariant and lattice homology; no free parameters or new entities are introduced.

axioms (1)

domain assumption Zemke's isomorphism identifies the relevant lattice homology groups with the Heegaard Floer groups and preserves the absolute grading used to define the d-invariant.
Invoked to transfer the lattice homology computation directly to the d-invariant.

pith-pipeline@v0.9.0 · 5388 in / 1256 out tokens · 60458 ms · 2026-05-08T15:34:57.759087+00:00 · methodology

discussion (0)

Forward citations

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Reference graph

Works this paper leans on

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Andr ´as N ´emethi,On the Ozsv´ ath-Szab´ o invariant of negative definite plumbed 3-manifolds, Geom. Topol.9(2005), 991–1042. MR2140997

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Stipsicz, and Zolt ´an Szab ´o,Knots in lattice homology, Comment

Peter Ozsv ´ath, Andr ´as I. Stipsicz, and Zolt ´an Szab ´o,Knots in lattice homology, Comment. Math. Helv.89(2014), no. 4, 783–818. MR3284295

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2003

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,The equivalence of lattice and Heegaard Floer homology, Duke Math. J.174(2025), no. 5, 857–910. MR4905538

2025