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Twisting, Stabilization and Bordered Floer homology

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves that every twist family with nonzero winding number eventually has knot Floer homology of fixed linear growth, with tau, thickness, and extremal invariants stabilizing under twisting.

desk verdict New linearity and stabilization results for HFK under nonzero-winding twisting, but the proof leans on Rasmussen's unpublished immersed curve machinery and two internal gaps leave the thickness theorem incomplete. read the letter →

arxiv 2507.15144 v1 pith:BROXW25U submitted 2025-07-20 math.GT

Soheil Azarpendar This is my paper · ORCID
classification math.GT MSC 57K1057R58
keywords twistfamilyknotFloerhomologyborderedimmersedcurveinvariantstabilizationAlexanderpolynomialtauthickness
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 studies what happens to a knot $K$ in $S^3$ when it is repeatedly twisted along a disjoint unknot $c$, equivalently when $-1/m$ surgery is performed on $c$. Its central claim is that for any such twist family $\{K_m\}$ with nonzero winding number, the knot Floer homology (a graded vector-space invariant of a knot) eventually changes in a fully controlled way: for all sufficiently large $m$, the total dimension, the $\tau$ invariant, and the thickness are affine functions of $m$, while the extremal knot Floer homology and the extremal Alexander coefficients stabilize. This matters because it reduces an infinite family of surgery problems to finitely many computations, and it removes the coherence assumption that earlier stabilization results required.

What carries the argument

The argument is carried out in bordered Floer homology, an invariant of manifolds with torus boundary, by viewing $K_m$ as a gluing of two bordered three-manifolds and studying the box tensor product $\widehat{CFA}(H_K,z,w) \boxtimes \widehat{CFD}(H'_{1/m},z')$. Its basis is arranged around a circle as a central black box $C^\bullet$ together with $m$ white boxes, and two nested inclusion maps $\Phi_m$, $\Phi'_m$ let the complexes for different $m$ be compared. The load-bearing mechanism is that all minimal cycles and minimal relations have length bounded independent of $m$, and that away from a fixed closed ball around the black box the differential is equivariant under shift maps $R_\pm$; consequently every piece of data that controls extremal knot Floer homology, the Alexander jump sequence, $\tau$, and thickness is forced into a fixed finite ball, where it must stabilize as $m$ grows.

What would settle it

Take an explicit knot pattern in the solid torus with nonzero winding, such as the Mazur pattern worked out in the paper, and compute $\widehat{HFK}(K_m)$ for large $m$; if the dimension is not eventually $D|m|-d$ for fixed integers, or if the extremal Maslov shift between $K_m$ and $K_{m+1}$ is not the stated $F_K$, the central claim collapses.

Watch

Extended reading notes

Core claim

The paper's central discovery is that for a twist family $\{K_m\}$ with $\mathrm{wind}_c(K) \neq 0$, the extremal pieces of knot Floer homology are eventually periodic up to a fixed Maslov shift: $\widehat{HFK}(K_m, -g(K_m)+j) \cong \widehat{HFK}(K_{m+1}, -g(K_{m+1})+j)[F_K]$ for each fixed $j$ and all large $m$, where $[\cdot]$ decreases the Maslov grading by $F_K$. From this single stabilization statement the paper derives the linear growth formulas $\dim \widehat{HFK}(K_m) = D|m|-d$, $\tau(K_m)=Tm+t$, and $\mathrm{th}(K_m)=Wm+w$, together with stabilization of the lowest Alexander coefficients and of the number of nonzero coefficient jumps. The shift $F_K$ is computed explicitly from the linking number of $K$ with $c$ and the Thurston norm of a planar surface in the complement, and it vanishes when $c$ links $K$ coherently.

Load-bearing premise

The argument depends on a set of announced but not yet published formulas about how knot Floer homology in a solid torus changes under a Dehn twist; if those formulas fail, the main theorems do not follow.

Editorial extensions

If this is right

  • For any fixed knot pattern with nonzero winding, the entire large-$m$ behaviour of $\widehat{HFK}(K_m)$ is determined by finitely many stabilized boxes; once the finite ball stabilizes, all larger $m$ are known.
  • The extremal $\widehat{HFK}$ stabilization means Alexander-grading-top properties of the knots $K_m$, such as concordance obstructions read from the top filtration, become periodic in $m$ up to the computable shift $F_K$.
  • The tau invariant and thickness being eventually affine in $m$ extends the earlier bounds for coherent twist families to every twist family with nonzero winding number.
  • The lowest-degree coefficients of the Alexander polynomial stabilize, and the coefficient sequence of $\Delta_{K_m}$ becomes a fixed series shifted by $t^{m\delta}$ with only finitely many positions where the coefficients still jump.
  • In the coherent case $F_K=0$, so the stabilized extremal knot Floer homology is exactly periodic as $m$ increases by one.

Reading between the lines

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

  • The slope $D$ in the dimension formula is, via the immersed-curve proof, an explicit sum of intersection numbers of the curve invariant of $K$ with lines of fixed slope; for a concrete pattern like the Mazur pattern worked out in the paper, one could compute $D$ directly and test the asymptotic formula in small examples.
  • The closed-ball-and-shift mechanism looks general: any family of bordered gluing problems where the type-D factor grows linearly and the differential is shift-equivariant away from a finite core should exhibit the same stabilization, suggesting a broader principle for surgery families.
  • The excluded zero-winding case is the natural next test; the paper's methods fail there because homogeneous cycles can spread arbitrarily far around the circle, so genuinely different asymptotic behaviour is plausible.
  • Because the core pairing and Dehn-twist formulas are cited to an unpublished source, the unconditional status of the theorems is tied to verification of that source; the rest of the paper provides a detailed conditional proof.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper studies a twist family {K_m} of knots obtained by performing (-1/m)-surgery on an unknot c, with K viewed as a knot in the complementary solid torus. Assuming nonzero winding number of K around c, it claims: total dimension of knot Floer homology grows linearly in |m| (Theorem 1.1); the extremal groups HFK(K_m, -g(K_m)+j) stabilize up to a computable Maslov shift (Theorem 1.2); extremal Alexander coefficients and the number of Alexander jumps stabilize (Theorems 1.3 and 1.4); and tau and thickness are eventually linear in m (Theorems 1.5 and 1.6). The proofs combine bordered Floer box tensor products, a detailed combinatorial analysis of the resulting chain complexes, and Rasmussen's unpublished immersed-curve invariant for knots in the solid torus.

Significance. If the external machinery is valid, the results are a substantial advance: they extend stabilization theorems from coherent twist families to all twist families with nonzero winding, compute the previously unknown Maslov shift in Theorem 1.2, and give the first linear-growth statement for total HFK dimension in this setting. The paper also contains useful explicit constructions: the type D invariants of the surgery solid tori, the box tensor product basis with its nested inclusions, and a series of combinatorial lemmas bounding the length of minimal cycles and relations. However, the central argument rests on two results of Rasmussen that are stated without proof and are unpublished, and one key combinatorial bound is explicitly left unproved. These dependencies make the paper, as it stands, a conditional proof rather than a self-contained one.

major comments (4)
  1. [§2.2, §3, Theorem 3.1] Theorem 3.1, the immersed-curve pairing theorem for Rasmussen's invariant hfk(Q), is stated without proof and attributed to the unpublished reference [Ras23]; the proof of Theorem 1.1 in Section 3 is a direct application of this result. Because [Ras23] is not publicly available, the linear-dimension claim (Theorem 1.1) is not verifiable from the present manuscript. The author should either include a complete proof, cite a publicly available version, or explicitly state Theorems 1.1-1.6 as conditional on this external result.
  2. [§3, proof of Theorem 1.1] Even granting Theorem 3.1, the dimensional computation invokes Theorem 2.6, which requires the immersed curves to be primitive and unobstructed and to be equipped with local systems. The proof reduces to primitive curves using Lemma 2.5, but it never checks that the components of hfk(K) are unobstructed, nor that the local system automorphisms satisfy the hypotheses of Theorem 2.6. If a component has a fishtail, the equality dim HF(ℓ_m, γ_j) = k_j · i(ℓ_m, γ_j) need not hold, so the constants D and d in Theorem 1.1 are not justified.
  3. [§4.5, Corollary 4.16; §8.2] Corollary 4.16, which bounds diam(∪_t I_t) for a minimal relation, is explicitly not proved in the text: the manuscript says 'We do not provide a full proof of Corollary 4.16.' This bound is load-bearing in Section 8: it is used in the proof of Proposition 8.2 to justify shifting a relation (Equation 8.1), and in Corollary 8.3 to show that the generating sets stabilize. Consequently, Theorem 1.6 is incomplete unless a proof of Corollary 4.16 is supplied.
  4. [§6.6, Theorem 1.3] The statement of Theorem 1.3 appears inconsistent with the proof in Section 6.6. There the author derives deg(Δ_{K_m}) = ((l/2) wind_c(K)) m + constant with 0 ≤ l ≤ x([D̂]), so the lowest-degree terms of Δ_{K_m} have a common shift of slope -(l/2) wind_c(K), not l/(2 wind_c(K)) as printed in the theorem. The sign and denominator in the displayed value of δ in Theorem 1.3 should be corrected to match the proof, and the interval for l should be reconciled with the sign convention used for 'first terms in increasing order of degree'.
minor comments (5)
  1. [§1.2] The sentence 'The first author [Aza23] showed...' should read 'the author', since the paper has a single author.
  2. [Abstract and Theorem 1.1] There are repeated typographical issues with the notation for HFK, e.g. '[HFK(K_m)' should be \widehat{HFK}(K_m), and 'dimension of [HFK(K_m) is given by' is missing a closing bracket.
  3. [§1.2] The phrase 'E := S3 − N (K ∪ c) repreresent the complement' contains a typo ('repreresent') and should be 'represents'.
  4. [Proposition 3.3] In the statement, 'the linear parts of τ∂D2(γ) comes from applying Dehn twist on the linear parts of τ∂D2(γ)' should refer to the linear parts of γ, not of τ∂D2(γ), on the second occurrence.
  5. [Lemma 4.2] The expression 'M ∈ Z/2' is ambiguous: later formulas such as β0 = M/2 + 1/4 indicate that M is intended to be a half-integer parameter, so the notation should be made precise.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the main results are derived from standard bordered Floer machinery and external (partly unpublished) immersed-curve theorems; the self-citation is background only.

full rationale

The derivation chain is not circular. Theorem 1.1 is proved by pairing the immersed-curve invariant hfk(K) with the explicitly computed invariant of H'_{1/m}, and the linear-dimension formula follows from HRW's intersection-number theorem (Theorem 2.6), Lemma 2.5, and the singular-pegboard computation, not from the conclusion being proved. Theorems 3.1 and 3.2 are imported from Rasmussen's unpublished work [Ras23], and the paper itself notes in Section 2.2 that 'Rasmussen's work on this invariant hasn't been published yet.' This is a genuine external-support gap, but it is not circular: the stated pairing and twist formulas are not defined in terms of the paper's target theorems, nor do the target theorems feed back into the hypotheses of those formulas. The later stabilization arguments for Theorems 1.2, 1.5, and 1.6 are developed combinatorially from the box tensor product via fixed-ball stabilization data; no fitted parameter is later renamed as a prediction. The self-citation [Aza23] in the introduction is contextual background and is not load-bearing. Section 6 is explicitly described as not containing new results ('While this section does not contain new results'), but that is disclosure of provenance, not circularity. The omitted proof of Corollary 4.16 ('We do not provide a full proof of Corollary 4.16') is a completeness gap affecting the Section 8 argument, but it is again not a definitional or self-referential reduction. On the quoted evidence, no circular step can be exhibited, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted numerical parameters and no new physical or topological entities. Its conclusions are derived from established bordered Floer machinery plus an unpublished immersed curve formalism by Rasmussen, which drives the correctness risk.

assumptions (5)
  • domain assumption Rasmussen's immersed curve invariant hfk for knots in a solid torus, with pairing theorem and Dehn twist formula.
    Stated as Theorems 3.1 and 3.2 in Section 3 and attributed to the unpublished reference [Ras23]; it is the foundation of the total dimension theorem.
  • standard math Bordered Floer pairing theorems of Lipshitz, Ozsvath and Thurston.
    Theorems 2.1 and 2.2 and the A-infinity invariance Theorem 2.3 are used throughout to identify the knot Floer complex of K_m with a box tensor product.
  • standard math Hanselman, Rasmussen and Watson immersed curve pairing and dimension formula.
    Theorems 2.4, 2.6 and 2.7 justify the intersection-number computation in Section 3.
  • domain assumption Baker-Taylor linear growth of Seifert genus under twisting.
    Theorem 1.7 from [BT16] is used in the proofs of Theorems 1.2, 1.3 and 1.5 to control the genus and the Maslov shift.
  • domain assumption Boundedness of the type A invariants of H_K, arranged up to homotopy.
    Assumed in Section 4 so that all higher operations m_n vanish for n at least L_A; the paper states this is not restrictive up to homotopy equivalence.

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Pith. "Pith review of Twisting, Stabilization and Bordered Floer homology." pith.science (2026). https://pith.science/paper/BROXW25U

@misc{pith2026250715144,
  author       = {Pith},
  title        = {Pith review of: Twisting, Stabilization and Bordered Floer homology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BROXW25U}},
  note         = {Machine review of arXiv:2507.15144}
}
abstract

Consider an unknot $c$ in $S^3$ and a knot $K$ in ${S^3-N(c)}$. Twisting the knot $K$ along $c$, or equivalently applying $\frac{1}{m}$-surgery on $c$, produces a family of knots $\{K_m\}_{m \in \mathbb{Z}}$. We use bordered Floer homology and the theory of immersed curve invariants to show that for $|m|\gg0$, total dimension of $\widehat{\mathrm{HFK}}(K_m)$, $\tau(K_{m})$ and thickness of $K_{m}$ are linear functions of $m$. Furthermore, we prove that the extremal coefficients of the Alexander polynomial and extremal knot Floer homologies of $K_m$ stabilize as $m$ goes to infinity. This generalizes results of Chen, Lambert-Cole, Roberts, Van Cott and the author on coherent twist families.

Figures

Figures reproduced from arXiv: 2507.15144 by the authors.

Figure 1
Figure 1. Quiver used to define torus algebra A [Che19] [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Decorated graph of a type D structure [HRW16] Higher-order δk are determined by directed paths in ΓN . The type D structure N is bounded if and only if ΓN contains no directed cycles. Simi￾larly, N is reduced if and only if ΓN has no edge with label ∅. A type A structure is defined as a right A∞-module over A. This means that a type A structure is a right unital I-module P, with a family of maps mi+1 : P ⊗A⊗i ! P, s… view at source ↗
Figure 3
Figure 3. Decorated graph of a type A structure [HRW16] A type A structure P and a type D structure N can be paired to form the box tensor product P ⊠ N. As a F2 vector space, P ⊠ N is isomorphic to P ⊗I N. The differential which turns P ⊠ N to a chain complex is defined as follows: (2.1) ∂ ⊠(x ⊗ y) = X∞ i=0 (mi+1 ⊗ IN )(x ⊗ δi(y)). Note that boundedness of N ensures that the sum in the above equation is finite, and therefore… view at source ↗
Figures from the paper (47 more)
Figure 4
Figure 4. Figure 4: Rules of grading a type D structure [HRW18] Defining the grading of a type A invariant CFD [(M, α1, α2) follows a similar procedure to that of type D. First, a reference generator must be chosen. Then, for any x, y ∈ CFA [(M, α1, α2) and a1 ⊗ · · · ⊗ an ∈ A⊗n , if y ap…
Figure 5
Figure 5. Figure 5: Rules of grading a type A structure [HRW18] For this paper, we work with knots in solid torus, and hence, the grading rules described are sufficient for determining the grading of all elements in the associated bordered invariants. To compute the Alexander factor of gr…
Figure 6
Figure 6. Figure 6: Embedding the decorated graph in T [HRW16] [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: Example of embedding a decorated graph as a train track [HRW16] Hanselman, Rasmussen and Watson [HRW16] also proved a pairing theo￾rem for the immersed curve invariants which we will recall in Theorem 2.4. Similar to before, assume that M and M′ are 3-manifolds with to…
Figure 8
Figure 8. Figure 8: The manifold Tϵ a geometric model for T. The figure om the right shows the projection p : Tϵ ! Tbϵ [HRW16] We can use the ϵ-geodesics to directly compute the intersection numbers, but we can also use them to construct the ϵ-pegboard diagrams. Given an ϵ-geodesic γ ⊂ Tϵ…
Figure 9
Figure 9. Figure 9: Turning a geodesic corner to a pegboard corner [HRW16] they are in minimal position. We can also go one step further and define a singular pegboard diagram. The main reference [HRW16] contains a more general definition, but for our purposes, we only need a specific cas…
Figure 10
Figure 10. Figure 10: Labeling arcs on the boundary of the genus-one bordered Heedaard diagram used to define type A invariants to closed curves in ∂M. We call a 4-tuple (Σ, α, β, z) with the aforementioned properties a pointed genus-one bordered Heegaard diagram. We can also define a doub…
Figure 11
Figure 11. Figure 11: Labeling arcs on the boundary of the genus-one bordered Heedaard diagram used to define type D invariants can be seen in [PITH_FULL_IMAGE:figures/full_fig_p029_11.png]
Figure 15
Figure 15. Figure 15: Note that although the convention of Hedden and Levine [HL12] is only described for reduced decorated graphs, it can be easily extended to non￾reduced graphs. For unreduced graphs, the maps mn+1 for n ≥ 1 are still defined by directed paths in the graph which doesn’t …
Figure 12
Figure 12. Figure 12: Mazur pattern in solid torus [PW20] [PITH_FULL_IMAGE:figures/full_fig_p030_12.png]
Figure 13
Figure 13. Figure 13: Doubly-pointed genus-one bordered Heegaard diagram associated to the Mazur pattern [PITH_FULL_IMAGE:figures/full_fig_p030_13.png]
Figure 14
Figure 14. Figure 14: Graphical representation of CFA [(HQ, z, w) [PITH_FULL_IMAGE:figures/full_fig_p030_14.png]
Figure 15
Figure 15. Figure 15: Graphical representation of CFA [(HQ, z) Now in order to find the grading structure of CFA [(HQ, z, w), we first need to pick an arbitrary reference generator. Set gr(y4) to be (0; 0, 0; 0). The [PITH_FULL_IMAGE:figures/full_fig_p030_15.png]
Figure 16
Figure 16. Figure 16: Index one embedded disks in Σe which doesn’t include lifts of z and w grading map will be : gr : CFA [(HQ, z, w) ! P(y4)\Ge Using the rules described in [PITH_FULL_IMAGE:figures/full_fig_p031_16.png]
Figure 17
Figure 17. Figure 17: Index one embedded disks in Σe which doesn’t include lifts of z, but include a lift of w As we mentioned P(y4) comes from the periodic domains in the diagram HQ. In our case, since the bordered 3-manifold described by the diagram [PITH_FULL_IMAGE:figures/full_fig_p03…
Figure 20
Figure 20. Figure 20: This diagram should be interpreted with care, as it omits the [PITH_FULL_IMAGE:figures/full_fig_p033_20.png]
Figure 18
Figure 18. Figure 18: The embedding of a decorated graph represent￾ing CFD [(HQ, z, w) in puntured torus T [PITH_FULL_IMAGE:figures/full_fig_p034_18.png]
Figure 19
Figure 19. Figure 19: Immersed curve invariant of Mazur pattern [PITH_FULL_IMAGE:figures/full_fig_p034_19.png]
Figure 20
Figure 20. Figure 20: A peg-board diagram of components of hfk(Q) (up to translation) Let’s denote this type D module as CFD [(H′ 1 m , z′ ) for m ∈ Z +. We start by computing CFD [(H′ 1 3 , z′ ). A bordered Heegaard diagram for this bordered manifold can be seen in [PITH_FULL_IMAGE:figur…
Figure 21
Figure 21. Figure 21: (H∞, z) which is the result of an isotopy on the β curve in (HQ, z) CFD [(H′ 1 3 , z′ ) · ι0 = ⟨η⟩, CFD [(H′ 1 3 , z′ ) · ι1 = ⟨ξ1, ξ2, ξ3⟩. And the δ map is as follows: δ(η) = ρ3 ⊗ ξ1 + ρ1 ⊗ ξ3, δ(ξi) = ρ23 ⊗ ξi+1 i = 1, 2. As a result CFD [(H′ 1 3 , z′ ) can be desc…
Figure 24
Figure 24. Figure 24 [PITH_FULL_IMAGE:figures/full_fig_p035_24.png]
Figure 22
Figure 22. Figure 22: Genus-one bordered Heegaard diagram (H′ 1 3 , z′ ) for H′ 1 3 We can now easily generalize this computation to CFD [(H′ 1 m , z′ ). The β curve in H′ 1 m intersects α2 in one point and α1 in m points and hence: CFD [(H′ 1 m , z′ ) = ⟨η, ξ1, · · · , ξm⟩F2 , [PITH_FULL…
Figure 23
Figure 23. Figure 23: Index one embedded disks in Σe counted in def￾inition of CFD [(H′ 1 3 , z′ ) [PITH_FULL_IMAGE:figures/full_fig_p036_23.png]
Figure 24
Figure 24. Figure 24: Decorated graph representing CFD [(H′ 1 3 , z′ ) CFD [(H′ 1 m , z′ ) · ι0 = ⟨η⟩, CFD [(H′ 1 m , z′ ) · ι1 = ⟨ξ1, · · · , ξm⟩. The count of index one disks used in definition of map δ is also very similar δ(η) = ρ3 ⊗ ξ1 + ρ1 ⊗ ξm, δ(ξi) = ρ23 ⊗ ξi+1 i = 1, · · · , m − …
Figure 25
Figure 25. Figure 25: Decorated graph representing CFD [(H′ 1 m , z′ ) and for i = 1, · · · , m: grm(ξi) = (1 2 ; 0, 1) · grm(ξi+1) ⇒ grm(ξi) = (− 1 2 ; 1 2 , − 2i − 1 2 ). Also we can compute P(η) as follows: grm(η) = (1 2 ; 1 2 , − 1 2 ) · grm(ξm) ⇒ grm(η) = (− m − 1 2 ; 1, −m). This mea…
Figure 26
Figure 26. Figure 26: The embedding of a decorated graph represent￾ing CFD [(H′ 1 3 , z′ ) in puntured torus T Remark 2.9. In Section 4, we examine CFD [(H′ 1 m , z′ ) when m ! ∞. More specifically we will compare the complexes CFD [(H′ 1 m , z′ ) and CFD [(H′ 1 m+1 , z′ ). Going on, abusi…
Figure 28
Figure 28. Figure 28: γ and meridional lifts Mn We can see two more complicated examples of meridian shifting in the Figures 31 and 32. In these examples we perturbed the horizontal segments a bit to keep the diagrams immersed. We can first show that that if we apply the meridian shifting …
Figure 29
Figure 29. Figure 29: Meridian shifting construction [PITH_FULL_IMAGE:figures/full_fig_p043_29.png]
Figure 31
Figure 31. Figure 31: Another example of meridian shifting (left to right) The homotopy is through the bigons in plane bound by L ′ i and τ∂D2 (Li). We only need to show there is no pegs inside these bigons. This follows from a simple geometric argument sketched in Figures 33 and 34. Now w…
Figure 32
Figure 32. Figure 32: Another example of meridian shifting (left to right) [PITH_FULL_IMAGE:figures/full_fig_p044_32.png]
Figure 34
Figure 34. Figure 34: Bigons don’t contain the punctures We call this line Hyi . If Li and Li+1 are one the same side of Hyi , the corner can’t be unwrapped. Without loss of generality assume they are both above Hyi . Note that L ′ i and L ′ i+1 remain above Hyi , while for an unwrapping t…
Figure 35
Figure 35. Figure 35: Only configuration for an unwrapping when Li and Li+1 are both above Hyi . This can’t happen in meridian shifting. Case 1: When θi+1 > θi : Note that unwrapping can only happen when θ ′ i+1 < θ′ i as seen in [PITH_FULL_IMAGE:figures/full_fig_p045_35.png]
Figure 36
Figure 36. Figure 36: Only configuration for an unwrapping in Case 1. As discussed above, this can’t happen in meridian shifting. Case 2: When θi+1 < θi : This case can only happen when the curve wraps around the peg and self￾intersects near ci . Otherwise the corner ci can be discarded wi…
Figure 38
Figure 38. Figure 38: L ′ i , L′ i+1 in Case 2 □ Iterated application of Proposition 3.3 gives us the following corollary about the behaviour of the immersed curve invariant under twisting. Corollary 3.4. There exist N ∈ N such that for m ≥ N, the slopes of (linear segments in a singular p…
Figure 39
Figure 39. Figure 39: The circular arrangement of the boxes in Cm Note that both of the chain complexes have the same base vector space and only differ in their differential. Since we are working over F2, the base set is in one-to-one correspondence with 2 Cm as follows: I ⊆ C m ! SI = X ν…
Figure 40
Figure 40. Figure 40: The natural inclusion Φ ′ m [PITH_FULL_IMAGE:figures/full_fig_p049_40.png]
Figure 41
Figure 41. Figure 41: The inclusion Φm We now start analyzing the properties of these complexes. We start by examining the differentials in Proposition 4.1. We use the graph-theoretic description described earlier. Proposition 4.1. There are four types of directed edges in the graph￾theore…
Figure 42
Figure 42. Figure 42: The complex CFA [(HQ, z, w) ⊠ CFD [(H′ 1 3 , z′ ) 4.2. Gradings in CFAd (HK, z, w) ⊠ CFD \(H′ 1 m , z′ ). We need to start by examining the gradings. We start by Lemma 4.2 about the grading of CFA [(HK, z, w). Recall that the solid torus H is defined as S 3 \ N(c) for…
Figure 43
Figure 43. Figure 43: The complex CFA [(HQ, z) ⊠ CFD [(H′ 1 3 , z′ ) [PITH_FULL_IMAGE:figures/full_fig_p052_43.png]
Figure 44
Figure 44. Figure 44: The decomposition of edges to four types de￾scribed in Proposition 4.1 Proof. As mentioned in [HRW18], the SpinC component of the generator of P(z0) is determined by the topology of the bordered 3-manifold. This means that the SpinC component will be equal to the one …
Figure 45
Figure 45. Figure 45: Illustration of the proof of Lemma 4.10 4.5. Length and size of minimal relations. To understand the homology, we also need to examine some of the prop￾erties of boundaries. To this end, we define the space of relations. Definition 4.12. Let I1, · · · , Ir ⊆ Cm, and a…
Figure 46
Figure 46. Figure 46: Decorated graph representing CFA [(H∞, z) Lemma 4.18. Let fb be the special A∞ homotopy equivalence defined above. Consider a subset Ibm ⊆ Cm such that fb⊠ Im(x0 ⊗ η) = SIbm . Then the cycle SIbm is homogeneous with respect to grSm. Furthermore, the grading double cos…
Figure 47
Figure 47. Figure 47: A schematic picture of an alternating path [PITH_FULL_IMAGE:figures/full_fig_p072_47.png]
Figure 48
Figure 48. Figure 48: The closed metric ball B• t Proposition 5.1. For any subsets I ⊆ Cm \ B• LA+1, we have ∂SI = SJ ⇒ ∂SR±(I) = SR±(J) . Proof. First, note that due to Lemma 4.9, we have that J ⊆ C m \ B • 1 . As a result, R±(J) is well-defined. The proof follows from Proposition 4.1. Si…
Figure 49
Figure 49. Figure 49: Constructing Cm+1 from Cm by adding a new white box. For m > 2L, this process doesn’t affect B• L [PITH_FULL_IMAGE:figures/full_fig_p078_49.png]
Figure 50
Figure 50. Figure 50: The half-ball B• [0,L] and its stabilization under natural inclusion Φ ′ m 5.3. Proof of Theorem 1.2. Now we are ready to prove Theorem 1.2. Proof of Theorem 1.2. Assume that m is high enough. To be precise assume that m > 2L where L := (LS + k − 1) + LB + 2LA [PITH_…
Figure 51
Figure 51. Figure 51: The subsets D′ m and D′′ m Now recall that HFK [(Km) = H∗(CFA [(HK, z, w) ⊠ CFD [(H′ 1 m , z′ )). For any integer i ∈ Z, we consider the subspace consisting of all homology classes in Alexander grading i with a representative sitting in D′ m i.e. Hi D′m := {H ∈ HFK [(…

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    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION format.date year duplicate empty "emp...

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