Lagrangian cobordisms and Legendrian invariants in knot Floer homology

C.-M. Michael Wong; John A. Baldwin; Tye Lidman

arxiv: 1907.09654 · v1 · pith:P7DOXZUInew · submitted 2019-07-23 · 🧮 math.GT · math.SG

Lagrangian cobordisms and Legendrian invariants in knot Floer homology

John A. Baldwin , Tye Lidman , C.-M. Michael Wong This is my paper

Pith reviewed 2026-05-24 17:28 UTC · model grok-4.3

classification 🧮 math.GT math.SG

keywords Legendrian linksLagrangian cobordismsknot Floer homologyLOSS invariantGRID invariantcontact geometrysymplectization

0 comments

The pith

LOSS and GRID invariants of Legendrian links behave functorially under decomposable Lagrangian cobordisms.

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

The paper shows that the LOSS and GRID invariants from knot Floer homology change in a controlled functorial manner when Legendrian links are joined by decomposable Lagrangian cobordisms in the symplectization of R^3. This controlled change produces explicit, computable obstructions that rule out the existence of such cobordisms between particular pairs of Legendrian links. A sympathetic reader cares because the obstructions are effective and arise directly from existing knot Floer homology calculations rather than from new geometric constructions.

Core claim

The LOSS and GRID invariants of Legendrian links in knot Floer homology behave in certain functorial ways with respect to decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on R^3. The results give new, computable, and effective obstructions to the existence of such cobordisms.

What carries the argument

Functoriality of the LOSS and GRID invariants under decomposable Lagrangian cobordisms.

If this is right

The invariants supply new obstructions to the existence of decomposable Lagrangian cobordisms between Legendrian links.
These obstructions can be checked by direct computation in knot Floer homology.
Certain pairs of Legendrian links are thereby shown to admit no such cobordism.

Where Pith is reading between the lines

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

The same functoriality might be checked for other knot Floer homology invariants of Legendrian links.
Explicit computations on standard families such as torus knots could produce concrete new examples of non-cobordant Legendrians.
If similar statements hold for non-decomposable cobordisms, the obstructions would apply more broadly.

Load-bearing premise

The cobordisms considered are restricted to decomposable Lagrangian cobordisms rather than arbitrary Lagrangian cobordisms.

What would settle it

A concrete decomposable Lagrangian cobordism between two Legendrian links for which the LOSS or GRID invariant at one end fails to determine the invariant at the other end in the predicted way would falsify the functoriality claim.

read the original abstract

We prove that the LOSS and GRID invariants of Legendrian links in knot Floer homology behave in certain functorial ways with respect to decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on $\mathbb{R}^3$. Our results give new, computable, and effective obstructions to the existence of such cobordisms.

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

The paper proves functoriality of the LOSS and GRID invariants under decomposable Lagrangian cobordisms, giving new computable obstructions.

read the letter

The main takeaway is that Baldwin, Lidman, and Wong prove the LOSS and GRID invariants in knot Floer homology are functorial with respect to decomposable Lagrangian cobordisms in the symplectization of the standard contact R^3. This produces new, computable obstructions to the existence of those cobordisms. What the paper does well is to make these invariants respect the cobordism structure in a way that lets you get effective information out of them. The focus on decomposable cobordisms keeps the constructions concrete and the results applicable to computations. The soft spots are limited. The restriction to decomposable cases is stated up front, so it matches the claim. There is no sign of circularity or invented entities. The work builds on prior definitions of the invariants without apparent fitting issues. If the proofs hold, the central argument is fine. This paper is for specialists in Legendrian knot theory and contact geometry who already use knot Floer homology. A reader who needs obstructions for specific cobordisms between Legendrian links will get direct value from it. I would bring this to a reading group on Floer homology topics. I would cite it in work that uses these obstructions. It deserves peer review because the result is a clear addition to the toolkit in the area.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that the LOSS and GRID invariants of Legendrian links in knot Floer homology are functorial with respect to decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on R^3. The results are presented as providing new, computable obstructions to the existence of such cobordisms.

Significance. If the stated functoriality holds, the work meaningfully extends the applicability of existing knot Floer homology invariants (LOSS and GRID) to questions about Lagrangian cobordisms, yielding effective obstructions in contact geometry. The explicit restriction of the claims to decomposable cobordisms is a strength, as it matches the scope of the theorem and avoids overclaiming.

minor comments (2)

The abstract states the main result clearly but does not indicate the precise form of the functoriality (e.g., whether the maps preserve the Alexander grading or how the invariants transform under the cobordism). Adding one sentence on this point would improve readability.
Notation for the invariants (LOSS, GRID) and the cobordism category should be introduced consistently in the introduction with forward references to the relevant sections where they are defined.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript. The report recommends minor revision but lists no specific major comments to address. We have therefore prepared no point-by-point responses and will incorporate any minor editorial suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity; theorem is self-contained

full rationale

The paper proves functoriality of the pre-existing LOSS and GRID invariants under a restricted class of decomposable Lagrangian cobordisms, as stated in the abstract. No load-bearing step reduces by definition, by fitted input, or by self-citation chain to the target claim itself; the derivation supplies new obstructions rather than renaming or re-deriving its inputs. The restriction to decomposable cobordisms is explicit and matches the claimed scope, leaving the central result independent of the paper's own fitted quantities or prior self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the prior construction of the LOSS and GRID invariants and on standard properties of knot Floer homology and Lagrangian cobordisms in symplectic geometry; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Standard properties of knot Floer homology and the LOSS/GRID invariants
The paper applies existing invariants whose definitions and basic properties are taken from prior literature.
domain assumption Decomposable Lagrangian cobordisms are well-defined objects in the symplectization of the standard contact R^3
The functoriality is stated specifically for this class of cobordisms.

pith-pipeline@v0.9.0 · 5580 in / 1323 out tokens · 41913 ms · 2026-05-24T17:28:01.269567+00:00 · methodology

Lagrangian cobordisms and Legendrian invariants in knot Floer homology

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)