Pith. sign in

REVIEW 2 major objections 6 minor 3 cited by

L-space satellite operators and knot Floer homology

T0 review · 2 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For any L-space pattern, the full knot Floer complex of a satellite is homotopy equivalent to a box tensor product built from the companion's knot Floer complex, making the satellite invariant algorithmically computable.

desk verdict Strong paper with a genuinely new formality theorem for 2-component L-space links, but the satellite formula for negative framings rests on an unproven truncation case that needs to be written out. read the letter →

arxiv 2412.05755 v1 pith:WTDWAFU4 submitted 2024-12-07 math.GT

Daren Chen , Ian Zemke , Hugo Zhou This is my paper · ORCID
classification math.GT MSC 57K1857K10
keywords knotFloerhomologyL-spacelinkssatelliteoperatorssurgeryalgebraformalityAlexanderpolynomialborderedHeegaardstaircasecomplexes
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper shows that for any L-space pattern $P$—meaning the two-component link formed by $P$ together with the core of the solid torus is an L-space link—the full knot Floer complex of the satellite $P(K,n)$ is determined by the knot Floer complex of the companion $K$. The formula is a homotopy equivalence expressing $\mathrm{CFK}(P(K,n))$ as a box tensor product of the surgery module of $K$, a Hopf-link bimodule, and a bimodule associated to the pattern link. The load-bearing step is a proof that every two-component L-space link has a formal link Floer complex, so the link Floer complex is determined by the multivariable Alexander polynomials of the link and its sublinks. Because cables, Whitehead doubles, and a family of Mazur patterns are L-space patterns, the paper turns satellite Floer computation into a finite algorithm, which the authors implement in Python.

What carries the argument

The machinery is the surgery algebra $\mathcal{K}$ from bordered reinterpretations of the link surgery formula, together with its type-D, type-A, and DA modules. The paper constructs a candidate bimodule $KY^{\Lambda}(L)_R$ from staircase complexes $C_s$ indexed by Alexander gradings, with module actions $L_W$, $L_Z$, $L_\sigma$, $L_\tau$ and higher homotopies chosen via the homological perturbation lemma. Koszul duality between the polynomial ring $F[W,Z]$ and the exterior algebra on two generators shows that the idempotent-$0$ part is a free resolution of the link Floer homology, and the vanishing lemma for $\operatorname{Hom}(S,S)$ in algebraic grading $-1$ and Maslov bigrading $(-1,-1)$ forces the candidate to match the genuine surgery bimodule. In the satellite formula, the pieces assemble into a $2\times 2$ hypercube of complexes $E,F,J,M$, with a truncation controlled by the companion's genus and the support of the pattern's $H$-function.

What would settle it

For a concrete two-component L-space link, compute the endomorphism space $\operatorname{Hom}(S,S)$ of its staircase complex in algebraic grading $-1$ and Maslov bigrading $(-1,-1)$; Lemma 5.19 predicts that the homology vanishes, and a nonzero class would falsify the formality theorem. Alternatively, run the paper's Python code on a specific L-space pattern and companion and independently recompute $\mathrm{CFK}(P(K,n))$ by another method; disagreement would falsify the satellite formula.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a reduction: the Floer-theoretic content of an L-space satellite operation is already encoded in the $H$-function of the pattern link, and hence in Alexander polynomial data. Theorem 1.2 states that $\mathrm{CFL}(L)$ is formal for every two-component L-space link $L$, meaning the chain complex is quasi-isomorphic to its homology; combined with the $H$-function description, this makes $\mathrm{CFL}(L)$ computable from Alexander polynomials. Theorem 1.5 extends the same conclusion to the surgery bimodule $KX^{\Lambda}(L)_R$, and the satellite formula then gives $\mathrm{CFK}(P(K,n))_R \simeq X_n(K)_K \boxtimes KH_K \boxtimes KX^{(0,0)}(L_P)_R$. The paper stresses that this computes the full complex over $F[W,Z]$, not merely its $U=0$ quotient, so surgery $d$-invariants and the $\Upsilon$ invariant become accessible.

Load-bearing premise

The two-component hypothesis is load-bearing: the proof that the candidate bimodule agrees with the genuine one uses Lemma 5.19, a vanishing statement about staircase endomorphisms that the paper shows fails for links with more than two components.

Editorial extensions

If this is right

  • For any L-space pattern, $\mathrm{CFK}(P(K,n))$ is computable from $\mathrm{CFK}(K)$ alone, with the computation implemented in Python.
  • The link Floer complex of a two-component L-space link is determined by the multivariable Alexander polynomials of the link and its sublinks (Corollary 1.3).
  • The formula covers all cabling operators, the Whitehead operator, generalized Mazur patterns, and the family of patterns built from pairs of L-space knots differing by a twist.
  • Because the full $F[W,Z]$-complex is computed, invariants that require more than the $U=0$ quotient—such as the $\Upsilon$ invariant and $d$-invariants of Dehn surgeries—become accessible.
  • The identity and elliptic involution cobordisms have the expected DA-bimodules, yielding a quasi-inverse for the algebraic bimodule that converts type-D to type-A modules.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Going beyond the paper: if formality held for $n$-component L-space links, the same surgery-algebra framework would give satellite formulas for patterns whose associated links have more components; the paper identifies the Ext vanishing that would have to be checked.
  • Going beyond the paper: the algorithmic nature of the formula suggests the satellite complex depends continuously on the companion's staircase data, so one could study how $\mathrm{CFK}(P(K,n))$ varies as $\mathrm{CFK}(K)$ is perturbed.
  • Going beyond the paper: the Python implementation makes it feasible to compute $d$-invariants of surgeries on iterated satellites, such as Whitehead doubles of cables, and compare them against known concordance obstructions.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper studies satellite operators P(K,n) for which the associated 2-component link LP is an L-space link, and gives a formula computing the full knot Floer complex CFK(P(K,n)) from CFK(K). The main tool is the link surgery formula reinterpreted via the surgery algebra K; the core theoretical step is a proof that 2-component L-space links have formal link Floer complexes (Theorem 1.2), which the authors prove using Koszul duality and staircase complexes. The authors then construct a candidate bimodule KY^Λ(L)_R, prove it is homotopy equivalent to the surgery bimodule KX^Λ(L)_R (Theorem 5.1), and use this to give a concrete model for the satellite complex as a box tensor product (Equation (1.1)). A substantial part of the paper is devoted to truncation of the infinite model to a finite one (Proposition 10.2) and to worked examples, including cables, Whitehead doubles, and Mazur patterns; the algorithm is implemented in Python.

Significance. If the main results are correct, this is a significant advance in knot Floer homology: it computes the full knot Floer complex of a large family of satellite knots from CFK(K) alone, going well beyond the U=0 or U-torsion information computed by previous bordered or immersed-curve techniques. The formality theorem for 2-component L-space links is itself an important structural result, and the paper provides many explicit calculations and reproducible Python code, which strengthens the value of the work. The proof strategies, especially the use of Koszul duality in Section 5, are original and likely to be influential.

major comments (2)
  1. [Section 10.2, Proposition 10.2] Proposition 10.2 states truncation results for all n ∈ Z, but the proof as written treats only n > 0 in detail. The cases n < 0 and n = 0 are dismissed with the sentence 'The arguments for the cases that n < 0 and n = 0 follow from very similar lines of reasoning. We leave the details to the reader.' This is a load-bearing gap: Equation (1.1) promises CFK(P(K,n)) for every integer framing, and the examples in Sections 11.3 (n = -1, 0) and 11.4 (n = -1) rely on the negative-framing truncation. The displayed truncation ranges in part (2) are not derived, and the filtration argument in Lemma 10.6, which uses that Φ_-K increases the s-filtration by n, does not directly apply when n < 0 because then Φ_-K decreases s and the left and right boundary contractions reverse roles. Please supply the missing details for the n < 0 and n = 0 cases before the main theorem can be accepted as fully proven.
  2. [Section 10 (Lemmas 10.4 and 10.5) and Section 7.2.3 (Lemma 7.10)] The proofs of two lemmas that are used in the central arguments are explicitly left to the reader. Lemma 10.5, whose proof is left to the reader, is needed in the proof of Proposition 10.2 to show that the boundary complexes C+ and C- are contractible; without this lemma the truncation argument for all framings is incomplete. Similarly, Lemma 7.10 is used in the computation of the elliptic bimodule K[E]_K (Theorem 1.6), which in turn is used in the proof of Theorem 5.1 through Corollary 7.13; its proof is also left to the reader. While these statements are plausible by analogy with the preceding lemmas, they are load-bearing for the main theorems, so the paper should either prove them or give a precise description of how the earlier arguments are modified.
minor comments (6)
  1. [Section 3.2, Lemma 3.7] The proof of Lemma 3.7 is left to the reader; since this is a standard homological perturbation lemma, the omission is acceptable, but a reference such as [HK91] should be explicitly cited in the lemma statement.
  2. [Section 1.2, Equation (1.1)] The notation CFK(P(K,n))_R is used in Equation (1.1) without a definition; please clarify in the introduction that the subscript R indicates the type-D module over R = F[W,Z] underlying the knot Floer complex.
  3. [Section 4.6] There is a typo at the end of the first paragraph of Section 4.6: 'XnY, K)' should be 'Xn(Y, K)'.
  4. [Throughout] The phrase 'Mauer-Cartan' (in Lemma 3.7 and the surrounding text) should be spelled 'Maurer-Cartan'.
  5. [Figures] The figures (e.g., Figures 6.1, 6.3, 7.2) are dense and difficult to read; indicating in the captions the conventions for distinguishing δ^1_1, δ^1_2, and δ^1_3 arrows would substantially improve readability.
  6. [References] The Python code is cited as [CZZ24]; please provide a persistent identifier or URL in the references so that the implementation can be located.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main computations are self-contained and the cited inputs are independent prior results.

full rationale

The paper's derivation chain does not reduce any prediction to its inputs by construction. Theorem 1.2 (formality of 2-component L-space link Floer complexes) is proved using Koszul duality and Lemma 5.19, an independent algebraic statement about staircase complexes; the candidate bimodule KY^Λ(L)_R is constructed from the H-function and then shown homotopy equivalent to KX^Λ(L)_R, not assumed equivalent. The H-function itself is imported from Gorsky–Némethi [GN16, Theorem 2.10] as an external formula from Alexander polynomials, with no parameter fitted to the output knots. The satellite formula (1.1) is assembled from the surgery/pairing formulas and the computed pattern bimodules; the examples in Sections 11 follow by explicit homological perturbation and truncation, with no fitted input masquerading as a prediction. The explicitly deferred n<0 case of Proposition 10.2 ('We leave the details to the reader') is a completeness gap, not a circular step, since the claimed reduction is still a theorem to be checked rather than an input. Self-citations to the second author's surgery algebra framework [Zem21a, Zem23] are prior established results with stated assumptions and do not smuggle in the target formality or satellite-formula conclusions. No specific identity of the form 'prediction equals fit by definition' could be exhibited, so the appropriate finding is no circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the surgery algebra framework of Zemke, the Gorsky-Nemethi H-function formula, and standard homological algebra. There are no fitted free parameters; the only notable modeling choice is the chiral topology on the surgery algebra, which the authors flag.

assumptions (6)
  • standard math Koszul duality between the polynomial ring and exterior algebra (Lemma 5.11).
    Used in Section 5.9 to prove formality of RY^0(L)_R by comparing tensor products with the dualizing bimodule RΛR.
  • standard math Homological perturbation lemma for A∞-modules and hypercubes (Lemmas 3.6, 3.7, 3.13).
    Used throughout to transfer structures, cancel contractible complexes, and construct truncations (e.g., Proposition 10.2).
  • domain assumption The surgery algebra K and the link surgery formalism of Zemke (Zem21a, Zem23), including the box tensor product formulas (Theorems 4.10, 4.11).
    The satellite formula (Equation 1.1) is derived in this framework; the paper cites prior work for the existence and invariance of the modules X_n(K)_K and KX^Λ(L)_K.
  • domain assumption Gorsky-Nemethi formula: for an L-space link, the H-function is determined by the multivariable Alexander polynomials of the link and its sublinks (Proposition 4.7).
    Used to compute the candidate bimodule KY^Λ(L)_R from Alexander data, and to justify Remark 10.3 that N is the highest power of t1 in the Alexander polynomial.
  • domain assumption Sublinks of L-space links are L-space links (Lemma 4.4, cited to Liu 2017).
    Used in Proposition 5.21 to conclude the second component of a 2-component L-space link is an L-space knot, so CFK(K2) is a staircase.
  • ad hoc to paper The results are stated in the chiral topology on the surgery algebra, not the U-adic topology (Remark 1.8).
    Theorem 1.6 and the truncation convergence arguments rely on the chiral topology; the authors explicitly note the U-adic version behaves differently (Remark 7.5).

how reviews work

0 comments
Cite this review

Pith. "Pith review of L-space satellite operators and knot Floer homology." pith.science (2026). https://pith.science/paper/WTDWAFU4

@misc{pith2026241205755,
  author       = {Pith},
  title        = {Pith review of: L-space satellite operators and knot Floer homology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WTDWAFU4}},
  note         = {Machine review of arXiv:2412.05755}
}
abstract

We consider satellite operators where the corresponding 2-component link is an L-space link. This family includes many commonly studied satellite operators, including cabling operators, the Whitehead operator, and a family of Mazur operators. We give a formula which computes the knot Floer complex of a satellite of $K$ in terms of the knot Floer complex of $K$. Our main tools are the Heegaard Floer Dehn surgery formulas and their refinements. A key step in our computation is a proof that 2-component L-space links have formal knot Floer complexes. We use this to show that the link Floer complexes of 2-component L-space links are determined by their multivariable Alexander polynomials. We implement our satellite formula in Python code, which we also make available.

Figures

Figures reproduced from arXiv: 2412.05755 by the authors.

Figure 1.1
Figure 1.1. µ P LP [PITH_FULL_IMAGE:figures/full_fig_p003_1_1.png] view at source ↗
Figure 2.1
Figure 2.1. The knot P(K, n) in terms of Dehn surgery on K#H#LP . 3. Algebraic background 3.1. Type-D and A modules. We make use of Lipshitz, Ozsv´ath and Thurston’s formalism of type-D, type-A and type-DA modules [LOT15] [LOT18]. Definition 3.1. Suppose that A is an algebra over a ring k. A type-D module XA consists of a right k-module X equipped with a k-linear map δ 1 : X → X ⊗k A such that (IX ⊗ µ2) ◦ (δ 1 ⊗ IA) ◦ δ 1 = 0. … view at source ↗
Figure 3.1
Figure 3.1. The maps appearing in the homological perturbation lemma for A∞-modules. We note that the above homological perturbation results have a more basic formalism in terms of chain complexes, which we now review. We will have use for this perspective. Lemma 3.7. Suppose that (C, d) is a chain complex and α is an endomorphism of C which satisfies the Mauer-Cartan relation α 2 + [d, α] = 0. (This implies that (C, d+α) is a … view at source ↗
Figures from the paper (44 more)
Figure 5.1
Figure 5.1. Figure 5.1: A staircase complex (left) and its homology (right). On the right, each dot represents a generator over F. The action of W shifts rightward, and the action of Z shifts upward. We now prove several straightforward lemmas which will help us later on. Lemma 5.6. Let A a…
Figure 6.1
Figure 6.1. Figure 6.1: Top right: the function HT2,6 . The boxes indicate the generators of type (y0-1) from Section 5.2. On the bottom is the bi￾module KX(0,0)(T2,6) F[W,Z] . The σ and τ arrows to the left and right of the region shown are weighted by either 1, W3U i or Z 3U i 6.2. The Wh…
Figure 6.2
Figure 6.2. Figure 6.2: The H-function and DA-bimodule for T2,2q. Similar as before, in the above diagram we record only δ 1 1 , as well as LW , LZ. We also have hZ,W (z− 1 2 ) = y− 1 2 ⊗ W, hW,Z(x 1 2 ) = y 1 2 ⊗ Z. Using the construction in Section 5.5, we obtain the full DA-bimodule stru…
Figure 6.3
Figure 6.3. Figure 6.3: The H-function and DA-bimodule of the positively clasped Whitehead link W+. In the diagram of the DA-bimodule (bottom diagram of [PITH_FULL_IMAGE:figures/full_fig_p042_6_3.png]
Figure 6.4
Figure 6.4. Figure 6.4: The H-function and DA-bimodule of the Mazur link M. Note that Cp,q(K, n) = Cp,q+pn(K, 0) C−p,−q(K, n) = Cp,q(rK, n) where rK denotes K with its string orientation reversed. Therefore for the sake of computing the cabling operators, it is sufficient to restrict to pos…
Figure 6.5
Figure 6.5. Figure 6.5: The two-bridge link K(6m+ 2, 3). The number is the box indicates the number of right-handed full-twists. The case when m = 3 is depicted. We can perform a similar procedure as before to obtain the H-function of Cp,q. From the H-function, we can compute the DA-bimodul…
Figure 6.6
Figure 6.6. Figure 6.6: The H-function and DA-bimodule of K(20, 3). δ 1 3 (σ, Z, x 0 − 3 2 ) = y 1 − 1 2 ⊗ W2 , δ1 3 (τ, W, z 0 3 2 ) = y 1 1 2 ⊗ Z 2 , δ 1 3 (σ, W, x 0 − 1 2 ) = y 1 − 3 2 ⊗ W2 , δ1 3 (τ, Z, x 0 1 2 ) = y 1 3 2 ⊗ Z 2 . 6.6. More general L-space patterns. We now describe som…
Figure 6.7
Figure 6.7. Figure 6.7: The H-function and DA-bimodule of the link C3,2 (the (3, 2)-cable of the Hopf link). Here, S denotes the the staircase complex of T2,3. There are also maps hW,Z, hZ,W , hσ,Z, hσ,W , hτ,Z and hτ,W which are not shown, but are described in Section 6.5 [PITH_FULL_IMAGE…
Figure 7.1
Figure 7.1. Figure 7.1: Representing the mapping cylinder of the identity map on the torus (left) and the elliptic involution (right) in terms of links. These bordered 3-manifolds are obtained by removing neighborhoods of the left and right most link components, and surgering on the middle.…
Figure 7.2
Figure 7.2. Figure 7.2: The DA-bimodule of the negative Hopf link KHK −. The gray arrows represent structure maps δ 1 1 : KHK − → KHK − ⊗ K. The remaining arrows are structure maps δ 1 2 : K ⊗ KHK − → KHK − ⊗ K. Sub￾scripts denote idempotents [PITH_FULL_IMAGE:figures/full_fig_p050_7_2.png]
Figure 8.1
Figure 8.1. Figure 8.1: The module KC F[W,Z] 2,1 . Only δ 1 1 and δ 1 2 (σ, −) and δ 1 2 (τ, −) are shown. Not shown are the maps δ 1 2 (W, −), δ 1 2 (Z, −) and δ 1 2 (T ±1 , −). There is no δ 1 3 [PITH_FULL_IMAGE:figures/full_fig_p059_8_1.png]
Figure 9.1
Figure 9.1. Figure 9.1: A model of the complex CFK(P(K, n)) given by the box tensor product X(P, K, n) F[W,Z] . 9.2. The complexes Es,t, Fs, Jt and M. We now describe the complexes appearing Es,t, Fs, Jt and M appearing in X(K, P, n). Note that we continue our assumption that Y is an intege…
Figure 10.1
Figure 10.1. Figure 10.1: The truncation from Proposition 10.2 when g = 1, N = 3 2 and n = −1. The truncation consists of the complexes in the region bounded by the blue line. Length 2 arrows are not shown. The squiggly red arrows are quasi-isomorphisms by Lemmas 10.4 and 10.5. − 5 2 − 5 2 −…
Figure 10.2
Figure 10.2. Figure 10.2: The truncation from Proposition 10.2 when g = 1, N = 3 2 and n = 1. The final result of the truncation process is the region bounded by the blue line [PITH_FULL_IMAGE:figures/full_fig_p074_10_2.png]
Figure 10.3
Figure 10.3. Figure 10.3: The truncation for g = 1, N = 3 2 , and n = −1. In the top right corner, Fg− 1 2 ,−N+ 1 2 Mg− 1 2 ,−N+ 1 2 ΦK is a contractible subcomplex of the truncation, and so is F−g+ 1 2 ,N− 1 2 M−g+n− 1 2 ,N− 3 2 Φ−K in the bottom left corner. Therefore, we can further trunc…
Figure 10.4
Figure 10.4. Figure 10.4: A further simplification for g = 1, N = 3 2 , and n = −1. When 0 ≤ n < 2g − 1, the parameter h = max {g, −g + n + 1} equals g. See [PITH_FULL_IMAGE:figures/full_fig_p081_10_4.png]
Figure 10.5
Figure 10.5. Figure 10.5: The truncation for g = 1, N = 3 2 , and n = 0 [PITH_FULL_IMAGE:figures/full_fig_p081_10_5.png]
Figure 10.6
Figure 10.6. Figure 10.6: A further simplification for g = 1, N = 3 2 , and n = 0. 10.2. Large framings. When n ≥ 2g−1, the truncation described in Proposition 10.2 has some useful properties. Note that when n ≥ 2g − 1, the ordinary mapping cone complex Xn(Y, K), which computes CF −(Yn(K)), …
Figure 10.7
Figure 10.7. Figure 10.7: The truncation for g = 1, N = 3 2 and the framing n = 2 In the bottom row, if n ≥ 2g, there is a contractible quotient complex of the form Eg+ 1 2 ,N− 1 2 Fg+ 1 2 ,N− 1 2 · · · E−g+n+ 1 2 ,N− 1 2 F−g+n+ 1 2 ,N− 1 2 that can be truncated, where the homotopy equivalen…
Figure 10.8
Figure 10.8. Figure 10.8: A further simplification for g = 1, N = 3 2 , and n = 2 F− 3 2 ,−1 E− 1 2 ,−1 F− 1 2 ,−1 E1 2 ,−1 F− 3 2 ,0 E− 1 2 ,0 F− 1 2 ,0 E1 2 ,0 F1 2 ,0 E− 1 2 ,1 F− 1 2 ,1 E1 2 ,1 F1 2 ,1 11. Examples In this section, we perform a number of example computations using the tr…
Figure 11.1
Figure 11.1. Figure 11.1: Es,− 1 2 for different values of s. Each • is a single gener￾ator. Similarly, Fs,− 1 2 is obtained by taking the first Alexander grading s part of the box tensor product of CFK(K) with F∗,− 1 2 , which is F∗,− 1 2 := · · · W2 01|T 0S W1 01|T 0S 101|T 0S Z 1 01|T 0S …
Figure 11.2
Figure 11.2. Figure 11.2: Fs for different values of s. Each ◦ is a single generator. The rightward horizontal map Φµ : Es,− 1 2 → Fs,− 1 2 is given by tensoring Xn(S 3 , K) K with the mapping cone of f µ . We note that in this case f µ acts by multiplication by W2 from C−1 to S, and multipl…
Figure 11.3
Figure 11.3. Figure 11.3: Φ K : Es,− 1 2 → Js,− 1 2 for different values of s. The map ΦK : Fs,− 1 2 → Ms,− 1 2 is obtained similarly by replacing δ 1 2 (σ, −) : E∗,− 1 2 → J∗,− 1 2 with δ 1 2 (σ, −) : F∗,− 1 2 → M∗,− 1 2 , which is F∗,− 1 2 : · · · S S S S S · · · M∗,− 1 2 : · · · S S S S S…
Figure 11.4
Figure 11.4. Figure 11.4: Φ −K : Fs, 1 2 → Ms+n,− 1 2 for different values of s. following diagram: Φ −K = Xn(K) · I0 ⊠ E 1 2 δ 1 1 (f −K) 1 2 Xn(K) · I1 ⊠ J− 1 2 ⊗ F[W, Z] τ We recall the map f −K is given by the diagram E∗, 1 2 : · · · C0 C0 C0 C1 C1 C1 · · · J∗,− 1 2 : · · · C0 C0 C0 C0 C…
Figure 11.5
Figure 11.5. Figure 11.5: Φ −K : Es, 1 2 → Js+n,− 1 2 for different values of s [PITH_FULL_IMAGE:figures/full_fig_p091_11_5.png]
Figure 11.6
Figure 11.6. Figure 11.6: Φ K : Fs,− 1 2 → Ms,− 1 2 for different values of s. There is no length 2 map in this example, since the actions of σ and τ commute with the actions of Z and W. Putting everything together, we obtain the truncated complex for the (2, 2n+1)-cable of the right hand tr…
Figure 11.7
Figure 11.7. Figure 11.7: A model of CFK(W+(T2,3, −1)), and a simplified version (3) When n = 1, CFK(W+(T2,3, 1)) is homotopy equivalent to the complex in [PITH_FULL_IMAGE:figures/full_fig_p100_11_7.png]
Figure 11.8
Figure 11.8. Figure 11.8: A model of CFK(W+(T2,3, 0)), and the simplified version The region F− 3 2 ,−1 E− 1 2 ,−1 F− 1 2 ,−1 E1 2 ,−1 M− 3 2 ,−1 J− 1 2 ,−1 M− 1 2 ,−1 J 1 2 ,−1 ΦK Φ−µ Φµ ΦK ΦK Φ−µ ΦK Φ−µ Φµ Φ−µ of the truncated complex, after similar simplification as before in the case of …
Figure 11.9
Figure 11.9. Figure 11.9: A model of CFK(W+(T2,3, 1)), and a simplified version • ◦ • • ◦ • • ◦ • Z W W Z UW UZ UW UZ W Z Z W • • • • • • • • • Z W Z Z Z W W W [PITH_FULL_IMAGE:figures/full_fig_p102_11_9.png]
Figure 11.10
Figure 11.10. Figure 11.10: A model of CFK(W+(T2,3, 2)), and a simplified version The information needed to build the lower half of the truncated complex can be obtained by the symmetry of rotation 180◦ about the plane, and then switching W with Z. We show some final results after some simpli…
Figure 11.11
Figure 11.11. Figure 11.11: A model of CFK(W+(T2,3, n)) when n ≥ 3 • ◦ • • ◦ • ⋆ ⋆ ∗ ⋆ ∗ ⋆ ⋆ ⋆ Z W U W Z W2 U U 1 W Z U U Z W Z U W Z W W2 U W [PITH_FULL_IMAGE:figures/full_fig_p103_11_11.png]
Figure 11.12
Figure 11.12. Figure 11.12: The region consisting of Es,−1, Fs−1,−1, Js,−1, Ms−1,−1 with s = − 1 2 , 1 2 for the Mazur pattern of the right hand trefoil ⋆ ⋆ ∗ ⋆ ∗ ⋆ ∗ ⋆ ⋆ • • ◦ • ◦ • ◦ • • Z W W Z Z W U W2 W Z U W2 Z Z U Z2 U U 1 W Z Z Z W W U W Z U W2 U Z U [PITH_FULL_IMAGE:figures/full_fig…
Figure 11.13
Figure 11.13. Figure 11.13: The region of interest Es,0, Fs−1,0, Js+n,−1, Ms+n−1,−1 for the Mazur pattern of the right hand trefoil. Note that J and M are independent of the s-coordinate. (1) When n = −1, CFK(Maz(T2,3, −1)) is homotopy equivalent to the complex in [PITH_FULL_IMAGE:figures/fu…
Figure 11.14
Figure 11.14. Figure 11.14: A model of CFK(Maz(T2,3, −1)). In the second dia￾gram, each vertical arrow is marked by Z and each horizontal arrow is marked by W. (4) When n ≥ 2, CFK(Maz(T2,3, n)) is homotopy equivalent to the one in [PITH_FULL_IMAGE:figures/full_fig_p104_11_14.png]
Figure 11.15
Figure 11.15. Figure 11.15: A model of CFK(Maz(T2,3, 0)). In the second diagram, each vertical arrow is marked by Z and each horizontal arrow is marked by W. the drawn one), and n − 2 many copies of the right square S2 on the right · · · . • • ◦ • ◦ ◦ • ◦ • • S1 S2 Z W U Z2 Z Z W W W2 U Z W …
Figure 11.16
Figure 11.16. Figure 11.16: A model of CFK(Maz(T2,3, 1)). In the second diagram, each vertical arrow is marked by Z and each horizontal arrow is marked by W. • • • • • ◦ • • ◦ • ◦ · · · ◦ • ◦ • ◦ · · · ◦ • ◦ • • ◦ • • • • • Z W Z U Z2 Z Z U Z2 Z W Z W Z U2W UW3 W W Z W2 U W Z W W W Z W2 U W Z…
Figure 11.17
Figure 11.17. Figure 11.17: A model of CFK(Maz(T2,3, n)) when n ≥ 2 [PITH_FULL_IMAGE:figures/full_fig_p106_11_17.png]
Figure 11.18
Figure 11.18. Figure 11.18: The E∗,t, F∗,t rows when the companion knot K is the right hand trefoil Then, we can remove arrows Ct− 1 2 1−→ Ct− 1 2 and Ct+ 1 2 1−→ Ct+ 1 2 , as they are contractible quotient complex of the row (and hence quotient complex of the whole 2−dimensional grid as well…
Figure 11.19
Figure 11.19. Figure 11.19: S C− 1 2 S C− 3 2 C− 1 2 S C− 1 2 C 1 2 S C− 1 2 C 1 2 S C 1 2 C 3 2 S C 1 2 S. Lτ Lσ U U Lτ hτ,Z LZ Lσ Lτ LW Lσ U hσ,W U U U Lτ hτ,Z LZ Lσ Lτ LW Lσ hσ,W U Lτ U Lσ [PITH_FULL_IMAGE:figures/full_fig_p108_11_19.png]
Figure 11.20
Figure 11.20. Figure 11.20: A model of CFK(C3,2(T2,3)). ◦ ◦ ⋆ ◦ ∗ ◦ ∗ ∗ ∗ ⋆ ◦ W Z3 UZ Z W2 U U Z2 Z W3 UW Z W W [PITH_FULL_IMAGE:figures/full_fig_p109_11_20.png]
Figure 11.21
Figure 11.21. Figure 11.21: A simplified model of CFK(C3,2(T2,3)) is replaced by the staircase Ct− 1 2 if A(x) ≥ s + 1 2 , and replaced by the staircase Ct+ 1 2 if A(x) < s + 1 2 , where A(x) is the Alexander grading of x. The maps in Es,t are such that if there is a differential y WiZj −−−−→…
Figure 11.22
Figure 11.22. Figure 11.22: Examples of Es,t and Fs,t when the input companion knot is an L-space knot and t = 1 2 . The red dashed line indicating where the Alexander grading equals s. The maps Φµ are replaced by Lσ, the maps Φ−µ are replaced by Lτ . The maps ΦK and Φ−K could be worked out s…
Figure 11.23
Figure 11.23. Figure 11.23: The E∗, 1 2 , F∗, 1 2 rows when the input is a square complex. After canceling contractible sub and quotient complexes, the row consisting of E∗,t and F∗,t is homotopy equivalent to the complex Ct shown in [PITH_FULL_IMAGE:figures/full_fig_p113_11_23.png]
Figure 11.24
Figure 11.24. Figure 11.24: The complex Ct , which is a simplification of the E∗,t, F∗,t rows when the input is a box complex. We observe that the complex in [PITH_FULL_IMAGE:figures/full_fig_p113_11_24.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Colored knot Floer homology: structures and examples

    math.GT 2025-08 conditional novelty 7.0 of 10

    The authors construct an n-colored knot Floer homology as a colimit over cable links with increasing full twists and equip it with a module structure over an explicit algebra.

  2. The link surgery formula and equivariant surgeries

    math.GT 2025-07 conditional novelty 7.0 of 10

    An equivariant link surgery formula is proved, giving diffeomorphism-induced maps on Heegaard Floer homology, and used to show the kernel of the forgetful map from the equivariant homology cobordism group contains a Z...

  3. Koszul duality and the link surgery formula

    math.GT 2025-07 conditional novelty 7.0 of 10

    The paper defines the curved dg-algebra K!, the Koszul dual of Zemke's surgery algebra K, and proves an equivalence between categories of bonsai, regularly U-adic modules over K and cobonsai, regularly U-adic type-D m...

Reference graph

Works this paper leans on

14 extracted references · 7 canonical work pages · cited by 3 Pith papers

  1. [7]

    L-SPACE SATELLITE OPERATORS AND KNOT FLOER HOMOLOGY 115 [HK91] Johannes Huebschmann and Tornike Kadeishvili, Small models for chain algebras , Math

    e-print, 2205.12798 [math.GT]. L-SPACE SATELLITE OPERATORS AND KNOT FLOER HOMOLOGY 115 [HK91] Johannes Huebschmann and Tornike Kadeishvili, Small models for chain algebras , Math. Z. 207 (1991), no. 2, 245–280. [Hog19] Matthew Hogancamp, Homological perturbation with curvature (2019). e-print, arXiv:1912. 03843. [Hom14a] Jennifer Hom, Bordered Heegaard Fl...

    arXiv 1991

  2. [8]

    Introduction to A-infinity algebras and modules

    [HR W24] Jonathan Hanselman, Jacob Rasmussen, and Liam Watson, Bordered Floer homology for manifolds with torus boundary via immersed curves , J. Amer. Math. Soc. 37 (2024), no. 2, 391–498. [HW23] Jonathan Hanselman and Liam Watson, Cabling in terms of immersed curves , Geom. Topol. 27 (2023), no. 3, 925–952. [Kel01] Bernhard Keller, Introduction to A-inf...

    work page 2024

  3. [13]

    [Zem21b] , The equivalence of lattice and Heegaard floer homology ,

    e-print, arXiv:2109.11520. [Zem21b] , The equivalence of lattice and Heegaard floer homology ,

    arXiv

  4. [14]

    e-print, arXiv:2308.15658. Department of Mathematics, California Institute of Technology, Pasadena, CA, USA Email address : darenc@caltech.edu Department of Mathematics, University of Oregon, Eugene, OR, USA Email address : izemke@uoregon.edu Department of Mathematics, University of Michigan, Ann Arbor, MI, USA Email address : hugozhou@umich.edu

    arXiv

  5. [183]

    [BG18] Maciej Borodzik and Eugene Gorsky, Immersed concordances of links and Heegaard Floer homology, Indiana Univ. Math. J. 67 (2018), no. 3, 1039–1083. [BLZ21] Maciej Borodzik, Beibei Liu, and Ian Zemke, Heegaard Floer homology, knotifications of links, and place curves with non-cuspidal singularities ,

    work page 2018

  6. [1969]

    Beilinson, Remarks on topological algebras, Mosc

    [Bei08] A. Beilinson, Remarks on topological algebras, Mosc. Math. J. 8 (2008), no. 1, 1–20,

    work page 2008

  7. [2003]

    L-spaces, taut foliations and the Whitehead link

    arXiv:math/ 0306378. [San22] Diego Santoro, L-spaces, taut foliations and the Whitehead link (2022). e-print, arXiv: 2201.01211. [Sei08] Paul Seidel, Fukaya categories and Picard-Lefschetz theory , Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Z¨ urich,

    work page Pith review arXiv 2022

  8. [2008]

    [Tur86] V. G. Turaev, Reidemeister torsion in knot theory, Uspekhi Mat. Nauk 41 (1986), no. 1(247), 97–147,

    work page 1986

Show all 14 references
  1. [2010]

    [Mot16] Kimihiko Motegi, L-space surgery and twisting operation , Algebraic & Geometric Topology 16 (2016), no

    e-print, arXiv:1011.1317. [Mot16] Kimihiko Motegi, L-space surgery and twisting operation , Algebraic & Geometric Topology 16 (2016), no. 3, 1727–1772. [OS03] Peter Ozsv´ ath and Zolt´ an Szab´ o,Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003), 615–639. [OS0...

    2016

  2. [2019]

    To appear in Geometry & Topology

    preprint, arXiv:1902.03333. To appear in Geometry & Topology. [GH17] Eugene Gorsky and Jennifer Hom, Cable links and L-space surgeries , Quantum Topol. 8 (2017), no. 4, 629–666. [GLM20] Eugene Gorsky, Beibei Liu, and Allison H. Moore, Surgery on links of linking number zero an...

    2017 arXiv

  3. [2020]

    [Mar01] Martin Markl, Ideal perturbation lemma, Comm

    e-print, arXiv:2009.05222. [Mar01] Martin Markl, Ideal perturbation lemma, Comm. Algebra 29 (2001), no. 11, 5209–5232. [MO10] Ciprian Manolescu and Peter S. Ozsv´ ath,Heegaard Floer homology and integer surgeries on links,

    2001

  4. [2021]

    To appear AGT

    e-print, arXiv:2104.13709. To appear AGT. [BLZ22] , Lattice homology, formality, and plumbed l-space links ,

    arXiv

  5. [2022]

    e-print, arXiv:2210. 15792. Accepted JEMS. [CH23] Wenzhao Chen and Jonathan Hanselman, Satellite knots and immersed Heegaard Floer homology (2023). e-print, arXiv:2309.12297. [Che19] Wenzhao Chen, Knot Floer homology of satellite knots with (1,1)-patterns (2019). e-print, arXi...

    2023 arXiv

  6. [2023]

    [Hed05] Matthew Hedden, On knot Floer homology and cabling , Algebr

    e-print, arXiv 2305.16271 [math.GT]. [Hed05] Matthew Hedden, On knot Floer homology and cabling , Algebr. Geom. Topol. 5 (2005), 1197–1222. [Hed07] , Knot Floer homology of Whitehead doubles , Geom. Topol. 11 (2007), 2277–2338. [Hed09] , On knot Floer homology and cabling. II ...

    2005 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.