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arxiv: 2605.18994 · v1 · pith:U2NDVURHnew · submitted 2026-05-18 · 🧮 math.GT · math.AG

Planar multilinks and rational singularities

M\'arton Beke , Olga Plamenevskaya This is my paper

Pith reviewed 2026-05-20 01:04 UTC · model grok-4.3

classification 🧮 math.GT math.AG
keywords surface singularitiesrational singularitiesplanar multilinksopen bookscontact structuressymplectic fillingsHeegaard Floer homology
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The pith

If the canonical contact structure on the link of a surface singularity is supported by a planar multilink open book, then the singularity must be rational.

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

The paper shows that surface singularities whose links carry the canonical contact structure via a planar multilink open book are necessarily rational. It also identifies sandwiched singularities as precisely those that admit planar multilinks containing a component of multiplicity one. Several properties known for ordinary planar open books extend directly to the multilink case: any symplectic filling is negative definite, no symplectic surface of positive genus can appear inside such a filling, and the Heegaard Floer contact invariant vanishes in the reduced group. The arguments combine existing results on fillings of planar spinal open books with lattice-embedding combinatorics.

Core claim

If the canonical contact structure on the link of a surface singularity is supported by a planar multilink open book, then the singularity must be rational. Sandwiched singularities are characterized by the existence of planar multilinks that include a component of multiplicity one. For any such link the symplectic fillings are negative definite and contain no symplectic surfaces of positive genus, while the image of the Heegaard Floer contact invariant vanishes in HF_red.

What carries the argument

Planar multilink open book that supports the canonical contact structure on the singularity link, together with lattice-embedding combinatorics.

If this is right

  • Symplectic fillings of the link are negative definite.
  • Such fillings contain no symplectic surfaces of positive genus.
  • The Heegaard Floer contact invariant vanishes in HF_red.
  • Sandwiched singularities admit planar multilinks with a multiplicity-one component.

Where Pith is reading between the lines

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

  • The criterion may give a contact-geometric test for rationality that applies to singularity links whose resolution graphs are not yet classified.
  • Similar restrictions might hold for Milnor fibers of higher-dimensional isolated singularities if planar multilink open books can be defined there.
  • One could test the claim by computing open-book decompositions for known non-rational singularities with small resolution graphs.

Load-bearing premise

Results on fillings of planar spinal open books extend without further restrictions to the multilink setting.

What would settle it

An explicit non-rational surface singularity whose link admits a planar multilink open book supporting the canonical contact structure.

Figures

Figures reproduced from arXiv: 2605.18994 by M\'arton Beke, Olga Plamenevskaya.

Figure 1
Figure 1. Figure 1: The graph D4 (left) and a sandwiched surface singularity resolution graph. any vertex is lowered. The graph on the right of [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left: the surface singularities Xn, where the long arm has n − 1 ≥ 1 vertices of self-intersection −2, not counting the node. Right: a factorization of their filling, with the vanishing cycles in red and the vanishing arcs in green. Note that only the last inner boundary component is encircled by two parallel vanishing cycles. Example 3.7. There is a double covering ∂A3 → ∂D4 induced by the inclusion Z4 ≤ … view at source ↗
Figure 3
Figure 3. Figure 3: The A3 and D4 graphs with the multiplicities and the strict transforms of the functions inducing the open books indicated. We need a generalization of another result of [GGP21], namely the calculation of the first Chern class c1(J) for an almost complex structure J compatible with the symplectic form ω on a symplectic filling (W, ω) of a planar multilink open book, given as a nearly Lefschetz fibration as … view at source ↗
Figure 4
Figure 4. Figure 4: A sandwiched graph and the corresponding planar multilink graph after capping off the outer boundary component. Unlabeled vertices are framed −2. By redrawing the diagram to take another boundary component to the outside, we recover [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
read the original abstract

Fibered multilinks are a generalization of classical fibered knots and open books that arise in the study of surface singularities and Milnor fibrations. We prove that if the canonical contact structure on the link of a surface singularity is supported by a planar multilink open book, then the singularity must be rational, and that sandwiched singularities are characterized by admitting planar multilinks with a component of multiplicity 1. We also show that some topological properties of planar open books extend to planar multilinks: symplectic fillings are negative definite and cannot contain symplectic surfaces of positive genus, and the image of the Heegaard Floer contact invariant vanishes in $HF_{red}$. Our results for singularities are based on these topological considerations, partly using Min--Roy--Wang's work on fillings of planar spinal open books, as well as the combinatorics of lattice embeddings.

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.

Referee Report

2 major / 2 minor

Summary. The paper proves that if the canonical contact structure on the link of a surface singularity is supported by a planar multilink open book, then the singularity is rational. It further characterizes sandwiched singularities as those admitting planar multilinks with at least one multiplicity-1 component. The authors establish that several properties of planar open books extend to the multilink setting: symplectic fillings are negative definite and contain no positive-genus symplectic surfaces, and the Heegaard Floer contact invariant vanishes in HF_red. These topological results are obtained via an extension of Min--Roy--Wang's work on planar spinal open books together with lattice-embedding combinatorics, and are then applied to deduce the rationality criterion.

Significance. If the central claims hold, the work supplies a contact-geometric and open-book criterion for rationality of surface singularities, extending classical results on Milnor fibrations and fibered links. The generalization of negative-definiteness, genus obstructions, and HF-vanishing from planar open books to planar multilinks is a useful technical contribution that may apply beyond singularity theory. The explicit use of Min--Roy--Wang together with lattice combinatorics provides a concrete bridge between symplectic fillings and singularity classification.

major comments (2)
  1. [§5, Theorem 5.1] §5, Theorem 5.1 (rationality implication): The proof that a planar multilink open book supporting the canonical contact structure forces the singularity to be rational rests on transferring negative-definiteness and the absence of positive-genus symplectic surfaces from Min--Roy--Wang's spinal-open-book results. The manuscript does not supply an explicit lemma verifying that the lattice-embedding constraints and spinal condition continue to hold when binding components carry multiplicities greater than 1; this step is load-bearing for the central claim.
  2. [§4, Proposition 4.5] §4, Proposition 4.5 (HF contact invariant): The vanishing of the image of the contact invariant in HF_red is asserted for planar multilinks by invoking the corresponding result for planar spinal open books. The argument does not detail how the canonical contact structure on the singularity link is preserved under the multilink binding with multiplicities, which is required to conclude that the vanishing implies rationality via the lattice combinatorics.
minor comments (2)
  1. [§2] The notation for multilink bindings and multiplicities is introduced in §2 but used without consistent reminders in later sections; adding a short table summarizing the binding data for the main examples would improve readability.
  2. Several citations to Min--Roy--Wang appear without page or theorem numbers; specifying the exact statements invoked would help readers trace the adaptations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below and will incorporate the suggested clarifications in a revised version.

read point-by-point responses

  1. Referee: [§5, Theorem 5.1] §5, Theorem 5.1 (rationality implication): The proof that a planar multilink open book supporting the canonical contact structure forces the singularity to be rational rests on transferring negative-definiteness and the absence of positive-genus symplectic surfaces from Min--Roy--Wang's spinal-open-book results. The manuscript does not supply an explicit lemma verifying that the lattice-embedding constraints and spinal condition continue to hold when binding components carry multiplicities greater than 1; this step is load-bearing for the central claim.

    Authors: We agree that an explicit verification would make the argument more transparent. The lattice-embedding constraints extend to multiplicities greater than 1 because the multiplicities appear as weights on the binding components in the intersection form, and the negative-definiteness and genus obstructions carry over directly from the spinal case via the same combinatorial counting of vertices and edges. To address the concern, we will insert a short lemma in Section 5 that records this extension explicitly, adapting the Min--Roy--Wang lattice arguments to the weighted multilink graph while preserving the spinal condition. revision: yes

  2. Referee: [§4, Proposition 4.5] §4, Proposition 4.5 (HF contact invariant): The vanishing of the image of the contact invariant in HF_red is asserted for planar multilinks by invoking the corresponding result for planar spinal open books. The argument does not detail how the canonical contact structure on the singularity link is preserved under the multilink binding with multiplicities, which is required to conclude that the vanishing implies rationality via the lattice combinatorics.

    Authors: The canonical contact structure on the singularity link is supported by any open book (including the multilink version) that realizes the Milnor fibration; the multiplicities correspond to the orders of the defining equations and do not alter the contact planes on the link. We will expand the paragraph preceding Proposition 4.5 with a brief remark recalling this standard fact from the theory of Milnor fibrations and open books for hypersurface singularities, thereby clarifying why the HF-vanishing result applies directly to the lattice-combinatorial criterion for rationality. revision: yes

Circularity Check

0 steps flagged

No circularity; proof extends external results to multilinks via cited theorems and combinatorics

full rationale

The paper derives its main theorems on rational singularities and sandwiched singularities from topological properties of planar multilink open books (negative-definiteness of fillings, absence of positive-genus symplectic surfaces, vanishing of the HF contact invariant) together with Min--Roy--Wang's results on planar spinal open books and lattice-embedding combinatorics. These steps invoke independent external theorems rather than reducing any claim to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. No equations or arguments in the provided abstract or description exhibit a derivation that is equivalent to its inputs by construction. The work is therefore self-contained against the cited external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The central claims rest on the extension of Min--Roy--Wang fillings to multilinks and on standard facts about canonical contact structures and lattice embeddings.

axioms (2)
  • domain assumption Min--Roy--Wang results on fillings of planar spinal open books extend to the multilink case without further restrictions
    Invoked to obtain negative-definiteness and vanishing of the contact invariant
  • domain assumption Lattice embeddings of the resolution graph control the existence of planar multilinks
    Used for the combinatorial part of the rationality and sandwiched characterizations

pith-pipeline@v0.9.0 · 5668 in / 1416 out tokens · 57131 ms · 2026-05-20T01:04:59.293064+00:00 · methodology

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