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arxiv: 2606.21834 · v1 · pith:YWNY7AOTnew · submitted 2026-06-20 · 🧮 math.GT

A note on the knot Floer homology of freely 2-periodic knots and their quotients

Timothy Bates , Aakash Parikh This is my paper

Pith reviewed 2026-06-26 11:23 UTC · model grok-4.3

classification 🧮 math.GT
keywords knot Floer homologyfreely periodic knotslocalization spectral sequenceSeifert genusquotient knotsgrid homology
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The pith

Freely 2-periodic knots have knot Floer homology rank at least as large as their quotients.

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

The paper establishes that if a knot P in the three-sphere is preserved setwise by a free order-two action, then the rank of its knot Floer homology is at least the rank of the knot Floer homology of the associated quotient knot. This inequality is derived by applying Large's generalization of the Seidel-Smith localization spectral sequence, which is originally defined for Lagrangian Floer homology, to the knot Floer setting. As a direct consequence, the Seifert genus of P is bounded below by the rational Seifert genus of the quotient. The authors also describe a computational tool that modifies an existing grid homology program to calculate the E2 page of the spectral sequence in concrete cases.

Core claim

For a freely 2-periodic knot P and its projective quotient knot Q, the rank of the knot Floer homology of P is greater than or equal to the rank of the knot Floer homology of Q. This follows from the existence of a spectral sequence arising from Large's generalization of the Seidel-Smith localization theorem for order-two actions.

What carries the argument

Large's generalization of the Seidel-Smith localization spectral sequence for order-two actions, applied to produce a rank inequality in knot Floer homology.

If this is right

  • The Seifert genus of the periodic knot is at least the rational Seifert genus of the quotient knot.
  • The E2 page of the spectral sequence can be computed explicitly for examples using a modified version of the Baldwin-Gillam grid homology program.
  • The rank inequality holds for any knot admitting a free order-two symmetry.

Where Pith is reading between the lines

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

  • The inequality supplies a new obstruction to a knot being the quotient of a freely 2-periodic knot.
  • Similar localization arguments might produce rank comparisons for other finite group actions on knots once the corresponding spectral sequences are available.
  • Explicit computations of the spectral sequence could reveal cases where the inequality is strict, giving quantitative information about the periodic action.

Load-bearing premise

Large's generalization of the Seidel-Smith localization spectral sequence applies directly to the knot Floer homology of freely 2-periodic knots in the three-sphere.

What would settle it

A concrete freely 2-periodic knot whose knot Floer homology has strictly smaller rank than the knot Floer homology of its quotient knot.

Figures

Figures reproduced from arXiv: 2606.21834 by Aakash Parikh, Timothy Bates.

Figure 1
Figure 1. Figure 1: Left: The freely 2-periodic knot P = 10157. The symmetry τ of P can be seen as rotation by 180 about the center dot followed by rotation by 180 in the direction of the blue curved arrow. Right: The quotient knot P ⊂ RP3 in the disk model for RP2 . Throughout this paper we will take P to be a knot in S 3 , P a class 1 knot in RP3 , and K a knot in an arbitrary closed three-manifold Y . Knot Floer homology i… view at source ↗
Figure 2
Figure 2. Figure 2: The placement of β curves near crossings [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A 4-pointed equivariant Heegaard diagram constructed from the free diagram ( [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A two-pointed Heegaard diagram for the quotient knot P ⊂ RP3 . The boundary components with the same labels are identified via the antipodal map. 3.2. The action on gradings. The equivariant Heegaard diagram H of Section 3.1 is invari￾ant under τ : S 3 → S 3 , and hence there is an induced set map on the knot Floer complexes τ# : CFK ^(H) → CFK ^(H). It should be noted that in general, the map τ# need not … view at source ↗
Figure 5
Figure 5. Figure 5: A four-pointed equivariant Heegaard diagram for the unknot a chain map, τe# is not necessarily chain homotopy equivalent to τ#. However, the manipulation of the Lagrangians described above to obtain a chain map τe# crucially does not affect the Alexander grading. Therefore Proposition 3.1 allows us to conclude that τe# preserves Alexander gradings. The spectral sequence of Theorem 4.4 is computed from (1) … view at source ↗
Figure 6
Figure 6. Figure 6: Left: An equivariant grid diagram for 12n403. Right: An RP3 grid diagram for the quotient of 12n403. 5.2. 12n403. Freely 2-periodic knots admit a rather simple type of equivariant Heegaard diagram: a grid diagram G in the sense of [MOS07, OSS15] with even grid size such that the first and third quadrants are marked in the same way (we will call these A boxes), and the second and fourth quadrants are marked… view at source ↗
read the original abstract

A knot P in the three-sphere is freely 2-periodic if it is preserved setwise by a free order-two action. There is a natural projective quotient knot associated to P. We establish a rank inequality between the knot Floer homologies of P and its quotient as a consequence of Large's generalization of Seidel--Smith's localization spectral sequence associated to order 2 actions in Lagrangian Floer homology. As a corollary we obtain an inequality between the Seifert genus of P and the rational Seifert genus of its quotient. We also implement a program which computes the E2 page of this spectral sequence using a modification of Baldwin--Gillam's grid homology calculator.

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

1 major / 0 minor

Summary. The manuscript claims to establish a rank inequality between the knot Floer homologies of a freely 2-periodic knot P in S^3 and its projective quotient knot, derived directly as a consequence of Large's generalization of the Seidel-Smith localization spectral sequence for order-2 actions in Lagrangian Floer homology. As a corollary it obtains an inequality relating the Seifert genus of P to the rational Seifert genus of the quotient. The authors also describe an implementation of a program computing the E2 page of the spectral sequence via a modification of Baldwin-Gillam's grid homology calculator.

Significance. If the application of Large's result is valid, the rank inequality supplies a new relation between Floer invariants of periodic knots and their quotients under free Z/2-actions, with potential utility for genus bounds and explicit computations. The accompanying computational tool adds practical value for verifying the spectral sequence in examples.

major comments (1)
  1. [Abstract] Abstract: the rank inequality is asserted to follow directly from Large's generalization, yet the manuscript supplies no explicit check that a free order-two action on a knot in S^3 induces an action on the underlying Heegaard or grid diagram data that satisfies all hypotheses of Large's theorem (e.g., commutation with the almost-complex structure, preservation of the necessary grading, and exact correspondence of the quotient complex to the invariant subcomplex).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying a point that improves the clarity of the argument. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the rank inequality is asserted to follow directly from Large's generalization, yet the manuscript supplies no explicit check that a free order-two action on a knot in S^3 induces an action on the underlying Heegaard or grid diagram data that satisfies all hypotheses of Large's theorem (e.g., commutation with the almost-complex structure, preservation of the necessary grading, and exact correspondence of the quotient complex to the invariant subcomplex).

    Authors: We agree that the manuscript would benefit from an explicit verification that the geometric hypotheses of Large's theorem are met. While the free Z/2-action on S^3 is by diffeomorphisms that can be arranged to preserve a suitable Heegaard diagram (or grid diagram) up to isotopy and to act compatibly with the almost-complex structure and gradings, the current text does not spell this out. In the revised version we will insert a short dedicated paragraph (likely in Section 2) confirming: (i) the action commutes with the almost-complex structure on the symmetric product, (ii) it preserves the relevant Maslov and Alexander gradings, and (iii) the quotient complex is precisely the invariant subcomplex. This addition will make the application fully rigorous without altering the main results. revision: yes

Circularity Check

0 steps flagged

No circularity; rank inequality follows from external theorem application.

full rationale

The paper's central claim is that the rank inequality is a consequence of Large's generalization of the Seidel-Smith spectral sequence. This is an external result with no overlap in authorship indicated. The derivation chain does not involve any self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations. The applicability to the specific setting is asserted but does not reduce the claimed inequality to a tautology by construction within the paper. No equations or ansatzes are smuggled via self-citation. This is a standard invocation of an independent theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of Large's cited generalization of the localization spectral sequence to this knot Floer setting and on the freeness of the order-two action.

axioms (1)
  • domain assumption Large's generalization of Seidel-Smith localization spectral sequence applies to knot Floer homology for free Z/2-actions on knots in S^3
    The inequality is explicitly a consequence of this generalization.

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