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arxiv: 2604.21240 · v1 · submitted 2026-04-23 · 🧮 math.GT

Recognition: unknown

Real link Floer homology

Yonghan Xiao

Pith reviewed 2026-05-08 13:37 UTC · model grok-4.3

classification 🧮 math.GT
keywords real link Floer homologystrongly invertible linksdoubly periodic linksreal grid diagramsHeegaard Floer homologyknot invariantssymmetric links
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The pith

Real link Floer homology is defined for strongly invertible and doubly periodic links in real 3-manifolds.

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

The paper defines a new homology theory for links that possess a strong inversion or double periodicity inside closed real three-manifolds whose fixed set is connected. This definition extends real Heegaard Floer homology from the case of manifolds or single knots to a broader class of links while preserving the real structure throughout the construction. In the special case of links inside the three-sphere, the theory is realized combinatorially through real grid diagrams, which replace the usual Heegaard surfaces with explicit diagrams that respect the involution. The resulting chain complex is used to extract structural information about the links and to compute concrete examples for many small strongly invertible knots. These calculations produce an appendix of data from which the author notes several recurring patterns.

Core claim

We define real link Floer homology for strongly invertible and doubly periodic links in closed real 3-manifolds with connected fixed sets, which generalizes real Heegaard Floer homology and real sutured Heegaard Floer homology. We give a combinatorial description of the theory in S^3 via real grid diagrams and use it to investigate structural properties of the theory as well as properties of strongly invertible knots.

What carries the argument

Real grid diagrams, which are grid presentations of links in S^3 that are equivariant under a fixed involution and generate a chain complex whose homology is the real link Floer homology.

If this is right

  • The homology supplies a combinatorial invariant for strongly invertible links in S^3 that can be calculated directly from grid diagrams.
  • The theory applies to all closed real 3-manifolds with connected fixed sets and recovers the earlier real Heegaard Floer theories in the appropriate special cases.
  • Explicit computations become feasible for small knots, producing concrete homology groups and revealing patterns in the data for more than fifty examples.
  • Structural properties such as behavior under connected sums or other operations on the links can be read off from the chain complex.

Where Pith is reading between the lines

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

  • The observed patterns in the computed examples for small knots may point to a relationship between the rank of the homology and the symmetry type of the knot.
  • Because the construction is combinatorial, it opens the possibility of machine-assisted searches for knots whose real link Floer homology detects previously invisible features of strong inversions.
  • The generalization to real 3-manifolds suggests the theory could be used to study equivariant problems in other Floer-type invariants that have real versions.

Load-bearing premise

The homology groups obtained from the construction do not depend on the choice of real Heegaard diagram or real grid diagram and remain unchanged when the link is modified by the allowed real moves.

What would settle it

Two different real grid diagrams for the same strongly invertible knot whose associated chain complexes have different homology groups, or a sequence of real link moves that alters the computed homology.

Figures

Figures reproduced from arXiv: 2604.21240 by Yonghan Xiao.

Figure 1
Figure 1. Figure 1: A genus one real Heegaard splitting of S 3 . 3. A combinatorial description In this section, we give a combinatorial description for the real link Floer homologies introduced in Section 2 following the strategy in [32]. We mainly focus on strongly invertible case, the periodic theory can be reformulated in exactly the same way. It is well-known that S 3 admits a unique real structure up to isotopy and any … view at source ↗
Figure 2
Figure 2. Figure 2: Choice of axis: in each frame, A is marked by a solid arrow and C ´ A is marked by a dotted arc view at source ↗
Figure 3
Figure 3. Figure 3: Involutive Reidemeister moves view at source ↗
Figure 4
Figure 4. Figure 4: An example of an I-move. ‚ real stabilization; ‚ real destabilization; ‚ real commutation. The definition of real grid moves follows immediately from the usual ones. More precisely, we perform a usual move with its reflection under R simultaneously. This is easy to imagine, so we decide not to provide details on this. The only case we need to remark is that we do not allow the stabilization to happen at an… view at source ↗
Figure 5
Figure 5. Figure 5: Realizing RI move as a real commutation view at source ↗
Figure 6
Figure 6. Figure 6: Example of real rectangles. real domains with nontrivial topology (annular and toroidal) and real index 1 (See [3, Sec￾tion 8.2]). Using this proposition, we can redefine the real link Floer homologies combinatorially as fol￾lows. Let H be a real grid diagram representing pL, aq, a strongly invertible link with auxil￾iary data. The O and X base points are labeled as in Subsection 2.1. Let RGpHq be the ring… view at source ↗
Figure 7
Figure 7. Figure 7: Choices of auxiliary data for the unknot. Note that modulo the equivalence relation of a, there are exactly four choices of a for a strongly invertible knot K Ă pS 3 , τ q- determined only by the choices (1), (2) in Definition 2.2. Fix any choice of a as a based data, the other three will be denoted by a r , a i and a i,r, in which r means orientation reversed while i means the role of X and O are intercha… view at source ↗
Figure 8
Figure 8. Figure 8: The result is the following: GHR zp¯31q “ Fp0,´1{2q ‘ Fp0,0q ‘ Fp1,1{2q GHR´p¯31q “ Frusp1,1{2q ‘ Fp0,0q . 21 view at source ↗
Figure 8
Figure 8. Figure 8: Grid diagrams for strongly invertible unknot, 31 and 41. GHR zp41q “ Fp´1,´1{2q ‘ Fp0,0q ‘ Fp0,1{2q GHR´p41q “ Frusp´1,´1{2q ‘ Fp0,1{2q . For ¯31, the rank of hat version coincides with the rank of usual knot Floer homology as predicted for L-space knots in [15]. In [15], Hendricks also calculated HFKR {p41q using spectral sequence from HFK to z HFKR (see Theorem 4.9), our result agrees with hers. Except t… view at source ↗
Figure 9
Figure 9. Figure 9: Equivariant crossing changes. Note that the conditions in Theorem 4.10 are still true for pM, L0, L1q “ pSymg pΣ ´ O ´ Xq, Tα, Tβq, when O, X base points on Σ are interchanged by R instead of lying on fixpRq. Thus, we have an analogue for doubly periodic knots. Theorem 4.11. For a doubly periodic knot pK, oq Ă pS 3 , τ q, there is a spectral sequence starting from HFK zpKq b Frθ, θ´1 s and converging to HF… view at source ↗
Figure 10
Figure 10. Figure 10: Grid diagrams for type A and B crossing change. where u 2 means that module action of multiplication by u 2 . When K` and K´ are related by a type B crossing change, then we have Frus-module maps C B ´ : GHR´pK`, a`q Ñ GHR´pK´, a´q, CB ` : GHR´pK´, a´q Ñ GHR´pK`, a`q of bigrading pMR, ARq “ p´1, ´1{2q, p´1, ´1{2q, respectively, so that C B ` ˝ C B ´ “ u 2 , CB ´ ˝ C B ` “ u 2 . In view at source ↗
Figure 11
Figure 11. Figure 11: Combined diagram for a type A crossing change. Recall from [32, Section 5.1], a pentagon in a grid diagram can be regarded as a rectangle with one edge replaced by a union of two consecutive segments in α` Y α´ (or β` Y β´) intersecting in a chosen distinguished vertex in α` X α´ (or β` X β´). Using this point of view, we can introduce a real pentagon as a real domain in a combined real grid diagram that … view at source ↗
Figure 12
Figure 12. Figure 12: Examples of pentagon and hexagon in a real grid diagram. H´px´q “ ÿ y´PpTα´ XTβ´ qR ÿ hPHexR,˝px´,y´q,hXX“H u nOf phq ź 1ďiďk U nOi phqy´; The calculation in [32, Lemma 6.2.1] works in real case without change, which shows that c A ˘ have the desired grading shifts. Note that the value of grading shift is actually the same- although real grading is roughly half of the original one, we perform a pair of cr… view at source ↗
Figure 13
Figure 13. Figure 13: Equivariant saddle moves. ‚ A pair of equivariant saddle moves can be thought of as attaching two bands to an equi￾variant Seifert surface of L equivariantly and taking the new boundary. In this case, the attaching region of the bands does not intersect the fixed points on L. lf is unaffected during this operation, while several different cases may happen to lp. (1) lp increases by 1; (2) lp decreases by … view at source ↗
Figure 14
Figure 14. Figure 14: Grid diagram for a paired saddle move. by σpxq “ # u 2 ¨ x if x P A x if x P B , and µpxq “ # x if x P A u 2 ¨ x if x P B . It is obvious that various compositions are all equal to the multiplication by u 2 . Now we check that they are chain maps: If x and y both belong to A or B, then it is obvious. Then the claim follows by noting that any real rectangle from x P A to y P B contains exactly one pair in … view at source ↗
Figure 15
Figure 15. Figure 15: Eliminate fixed 1-handles. After this, we take Ub1,b2 pK1q as f ´1 W p0.1q and Ud1,d2 pK2q as f ´1 W p1.9q. Since the cobordism is connected, for each of the b2 pairs of 2-component unlinks, up to some band slides along other pair of 1-handles, there must be a pair of bands (a pair of equivariant 1-handles) joining them to K1 or one of the d1 strongly invertible unknots. For each of the b2 pairs, we pick … view at source ↗
Figure 16
Figure 16. Figure 16: Canceling a fixed 0-1 handle pair view at source ↗
Figure 17
Figure 17. Figure 17: Eliminate fixed 1-handles from the handle decomposition of an equi￾variant slice surface of T2,3. Let A denote the fixed set in S, fixpτcq XS. By the equivariant assumption, a fixed critical point of h|S on A must also be a critical point of h|A and vice versa. Then our goal is to show that we can isotope S relative to its boundary to make h|A has a single local minimum, or equivalently, h|S has a single … view at source ↗
Figure 18
Figure 18. Figure 18: Real grid diagram for equivariant connected sums. Here, we add a subscript “std” to note that we only allow the standard involution on S 3 ˆ I and require the concordance to be smooth in contrast to the existing notion of concordance group, which allows any involution on S 3ˆI extending the standard one on ends and locally flat embedded annuli. We have to add restrictions due to the combinatorial and Mors… view at source ↗
Figure 19
Figure 19. Figure 19: Real oriented skein triple. this section, whenever we talk about a real oriented skein triple, we fix a triple of auxiliary data constructed in this way, and abuse the same notation a for all of them. When there is no ambiguity, this shall be omitted from the notation. Theorem 7.1. Let pL`, L´, L0, aq be a real oriented skein triple defined as above. ‚ If l 1 p “ lp ` 1, we have a long exact sequence (1) … view at source ↗
Figure 20
Figure 20. Figure 20: Grid diagrams for an real oriented skein triple. ‚ In H`, we use X` “ tX1, X1 1 , X2, X1 2 , . . .u and the pair of α and β circles colored in light blue and dark red. The corresponding families will be denoted α`, β`. ‚ In H´, we use X´ “ tXr1, Xr1 1 , X2, X1 2 , . . .u and the pair of α and β circles colored in purple and pink. The corresponding families will be denoted α´, β´. ‚ In H0, we use X0 “ tY1,… view at source ↗
Figure 21
Figure 21. Figure 21: Combined diagram for oriented skein relation. the following mapping cone expression for various real grid chain complexes. GHR´pH`q “ ConepBN` I` : I` Ñ N`q; GHR´pH0q “ ConepBI` N` : N` Ñ I`q; GHR´pH´q “ ConepBI´ N´ : N´ Ñ I´q; GHR´pH1 0 q “ ConepBN´ I´ : I´ Ñ N´q. The real grid states in H` and H0 can be identified naturally, and of course, the same is true for H´ and H1 0 . However, the real Alexander g… view at source ↗
Figure 22
Figure 22. Figure 22: Real unoriented skein triple. Proof of Theorem 7.2. The proof is almost identical to the previous one. We just need to note that in the hat chain complex, we just further let the variables that were set equal in the collapsed minus version to be all zero. □ 7.2. Decategorification. Definition 7.11. Let pL, aq be a generalized strongly invertible link with auxiliary data in S 3 . We define its real Alexand… view at source ↗
Figure 23
Figure 23. Figure 23: Loosing the restriction on auxiliary data. two gradings on H into a single δ R-grading defined by δ Rpxq “ MRpxq ´ A Rpxq, δRpuiq “ ´1 2 , δRpUj q “ ´1. Then, one see that δ Rpxq “ 1 2 pMR Opxq ` MR Xpxqq ` pn ´ lf ´ 2lpq 4 “ 1 4 pMOpxq ` MXpxqq ´ 1 4 |x X C| ` 1 4 lf ` pn ´ lf ´ 2lpq 4 Recall that the classical δ grading was defined by δpxq “ MOpxq ´ Apxq “ 1 2 pMOpxq ` MXpxqq ` pn ´ lf ´ 2lpq 2 , so δ R… view at source ↗
Figure 24
Figure 24. Figure 24: Grid diagrams for a real unoriented skein triple. Lemma 7.18. Let pL`, L´, L0, aq be a real oriented skein triple and take associated real grid diagrams H`, H´, H0 satisfying the consecutive rows(columns) assumption. Applying the previous discussion, we obtain maps Pr`,´ and Pr´,`. Then both compositions Pr`,´ ˝ Pr´,` and Pr´,` ˝ Pr`,´ are null-homotopic. An adaptation of proof of [32, Proposition 9.6.1] … view at source ↗
Figure 25
Figure 25. Figure 25: Involution on twist knots: The left picture shows the first family of involutions, the middle and right figures show the second family of involution on even and odd twist knots, respectively. Example 8.4. Besides the interesting phenomenon, there are also some strongly invertible knots that do not enjoy an interesting real knot Floer groups, i.e., HFKR˝ fails to tell them apart from the standard strongly … view at source ↗
Figure 27
Figure 27. Figure 27: A grid diagram for 62 with one of its a strongly inversion. Using the calculation from [10], Hendricks computed the differential for the top horizontal spec￾tral sequence: a ` x, b and c survives to HFKR. Combining this, one can see that the generator { survives to HFx pS 3 q and HFR zpS 3 , τ q are not the “same”, while we know the bottom horizontal arrow in just the identity map. This is also reflected … view at source ↗
read the original abstract

In this paper, we define real link Floer homology for strongly invertible and doubly periodic links in closed real $3$-manifolds with connected fixed sets, which generalizes real Heegaard Floer homology and real sutured Heegaard Floer homology. We give a combinatorial description of the theory in $S^3$ via real grid diagrams and use it to investigate structural properties of the theory as well as properties of strongly invertible knots. A computer implementation was written by Zhenkun Li. An appendix including real grid homology for 50+ small knots is made jointly by Zhenkun Li and the author, from which we observe several interesting phenomenon.

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 defines real link Floer homology for strongly invertible and doubly periodic links in closed real 3-manifolds with connected fixed sets, generalizing real Heegaard Floer homology and real sutured Heegaard Floer homology. It supplies a combinatorial description in S^3 via real grid diagrams, uses this to investigate structural properties of the theory and of strongly invertible knots, includes a computer implementation, and provides an appendix with real grid homology computations for over 50 small knots.

Significance. If the invariance under diagram changes is established, the construction would furnish a new Floer-theoretic invariant adapted to real symmetries, extending existing real Heegaard Floer theories to links and enabling computational study of strongly invertible knots. The combinatorial grid-diagram approach together with the explicit computer implementation and tabulated examples constitute concrete strengths that would make the invariant immediately usable for further calculations and conjectures.

major comments (2)
  1. [Section describing the combinatorial chain complex and its invariance (the section immediately following the definition)] The central claim that the homology is a link invariant rests on the assertion that the chain complex defined from a real grid diagram is independent of the choice of diagram. The text does not supply the required case-by-case verification that every real analogue of a grid move (real stabilizations, real commutations, and real Reidemeister-type moves compatible with the strong inversion or double periodicity) induces a chain homotopy equivalence; without these explicit homotopy equivalences the homology remains diagram-dependent rather than an invariant of the link.
  2. [Introduction and the section on the definition of real link Floer homology] The generalization from real Heegaard Floer homology to the link setting is stated in the abstract and introduction, yet the manuscript supplies no derivation steps showing how the real link complex reduces to the known real Heegaard or sutured complexes when the link is a knot or when the diagram is a Heegaard diagram; this reduction is load-bearing for the claim that the new theory is a genuine generalization.
minor comments (2)
  1. Notation for the real generators and the differential counting real holomorphic disks (or their combinatorial analogues) should be introduced with a single consistent table or list of symbols to avoid ambiguity when the same symbols appear in both the geometric and combinatorial descriptions.
  2. [Appendix] The appendix computations are valuable; each table should include a brief statement of the diagram used and the grading conventions so that readers can reproduce the entries from the combinatorial definition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The two major points raised concern the explicit verification of invariance under real grid moves and the detailed reduction to the known real Heegaard and sutured theories. We address each below and will revise the manuscript to strengthen these aspects while retaining the combinatorial and computational contributions.

read point-by-point responses

  1. Referee: [Section describing the combinatorial chain complex and its invariance (the section immediately following the definition)] The central claim that the homology is a link invariant rests on the assertion that the chain complex defined from a real grid diagram is independent of the choice of diagram. The text does not supply the required case-by-case verification that every real analogue of a grid move (real stabilizations, real commutations, and real Reidemeister-type moves compatible with the strong inversion or double periodicity) induces a chain homotopy equivalence; without these explicit homotopy equivalences the homology remains diagram-dependent rather than an invariant of the link.

    Authors: We agree that explicit verification of invariance is necessary to establish the homology as a true link invariant. The manuscript defines the real grid moves and asserts invariance by analogy with the non-real grid homology literature, but we acknowledge that the full case-by-case construction of chain homotopy equivalences (for real stabilizations, commutations, and symmetry-compatible Reidemeister moves) is not written out in detail. In the revised version we will insert a new subsection immediately after the definition of the chain complex. This subsection will construct the required homotopy equivalences explicitly, adapting the standard arguments from Manolescu–Ozsváth–Sarkar and subsequent grid-homology papers to the real symmetric setting while preserving the real involution on the complex. These additions will make the invariance rigorous without changing the overall combinatorial framework. revision: yes

  2. Referee: [Introduction and the section on the definition of real link Floer homology] The generalization from real Heegaard Floer homology to the link setting is stated in the abstract and introduction, yet the manuscript supplies no derivation steps showing how the real link complex reduces to the known real Heegaard or sutured complexes when the link is a knot or when the diagram is a Heegaard diagram; this reduction is load-bearing for the claim that the new theory is a genuine generalization.

    Authors: We appreciate the referee’s emphasis on making the generalization explicit. By construction, a real grid diagram for a strongly invertible knot specializes to a real Heegaard diagram of the knot complement (or the closed manifold), with generators, differentials, and gradings matching those of real Heegaard Floer homology; the sutured case follows similarly when the diagram is viewed as a sutured Heegaard diagram. To render this reduction transparent, we will add a short paragraph in the introduction and a dedicated remark in the definition section that spells out the correspondence between the real link complex and the existing real Heegaard/sutured complexes in the knot and Heegaard-diagram special cases. This clarification will be purely expository and will not alter the definitions or results. revision: yes

Circularity Check

0 steps flagged

No circularity: definition via real grid diagrams is independent of inputs.

full rationale

The paper defines real link Floer homology directly from real grid diagrams for strongly invertible and doubly periodic links, generalizing prior real Heegaard Floer theories without self-referential equations or fitted parameters renamed as predictions. No load-bearing step reduces the homology to its own construction by definition, self-citation chain, or ansatz smuggling. Invariance under diagram moves is a standard proof obligation in such combinatorial Floer theories and does not constitute circularity per the enumerated patterns. The appendix computations provide external verification content rather than tautological support. This is a normal non-finding for a definitional generalization paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the prior existence of real Heegaard Floer homology and real sutured Heegaard Floer homology; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Real Heegaard Floer homology and real sutured Heegaard Floer homology are well-defined and invariant as previously constructed.
    The new theory is stated to generalize these earlier constructions, so their validity is presupposed.

pith-pipeline@v0.9.0 · 5389 in / 1268 out tokens · 28169 ms · 2026-05-08T13:37:42.091974+00:00 · methodology

discussion (0)

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