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An unoriented skein exact triangle in unoriented link Floer homology

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Pith's one-line read The paper establishes an unoriented skein exact triangle in unoriented link Floer homology: for any unoriented skein triple La, Lb, Lc in a 3-manifold, the three homology groups fit into a cyclic exact triangle with maps induced by band…

desk verdict A credible and substantial new skein exact triangle for unoriented link Floer homology; the main risk is the unenumerated local polygon counts in Sections 7–8. read the letter →

arxiv 2501.01047 v2 pith:XZ4XSWME submitted 2025-01-02 math.GT

Gheehyun Nahm This is my paper · ORCID
classification math.GT MSC 57K1857K10
keywords unorientedlinkFloerhomologyHeegaardskeinexacttrianglebandmaps2-surgeryholomorphicpolygoncountsKhovanovlocalsystems
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's thesis is that unoriented link Floer homology, with two basepoints on each link component and coefficients in a power-series ring over $\mathbb{F}=\mathbb{Z}/2$, admits band maps that arrange any unoriented skein triple into an exact triangle. If this relation holds, the invariant behaves under the unoriented skein move the way equivariant Khovanov homology does, and it supplies the missing local step for a spectral sequence from Khovanov homology to knot Floer homology. The proof goes through a genus-one local computation and a Heegaard Floer analogue of a recent 2-surgery exact triangle in instanton Floer homology, relating unoriented knot Floer homology of $K$ to Heegaard Floer homology of two surgeries on $K$. A sympathetic reader would accept the main claim as the assertion that, in the local model, the only contributing holomorphic triangles, quadrilaterals, and pentagons are the ones explicitly listed and all others cancel.

What carries the argument

The load-bearing mechanism is a local computation on $T^2$ rather than on $Sym^2(T^2)$. The paper sets up a genus-one multi-Heegaard diagram with attaching curves $\beta_a,\beta_b,\beta_0,\beta_2,\tilde\beta_0,\tilde\beta_2,\beta_c$ and standard translates, with $\beta_c$ carrying a rank-two local system $E$ specified by an oriented arc $G$. The central relation is that the mapping cones $\beta_a\xrightarrow{\tau}\beta_b$ and $\beta_c^E$ are morally quasi-isomorphic, meaning there are cycles $\rho$ and $\sigma$ whose compositions with standard translates are $\Theta^+$ up to the unit $u=\sum_{n\ge1}U^{n^2-n}$ in the power-series coefficient ring. The proof is split into three claims showing that consecutive twisted complexes in the chain $\beta_a\xrightarrow{\tau}\beta_b$, $\beta_0\xrightarrow{\theta}\beta_2$, $\tilde\beta_0\xrightarrow{\tilde\theta}\tilde\beta_2$, $\beta_c^E$ are morally quasi-isomorphic, with the local counts reduced to triangle and quadrilateral counts in $T^2$. The band maps themselves are the composition maps $\mu_2(-,\Theta_B)$ for canonical cycles $\Theta_B$ in the Floer complex of the $\beta,\gamma$ curves, defined separately for non-orientable, split, and merge bands.

What would settle it

Compute the mod-2 counts of all Maslov-index 1, 2, and 3 polygons in the genus-one diagram of Figure 4.1 in a pinched almost complex structure, and check whether the resulting compositions $\mu_2(\rho,\sigma')$ and $\mu_2(\sigma,\rho')$ equal $\Theta^+$; any contribution outside the $T^\pm_n$ and $S^\pm_n$ families, or any nonzero correction term, would contradict Theorem 4.3 and break the triangle.

Watch

Extended reading notes

Core claim

The central claim is that unoriented link Floer homology satisfies the unoriented skein relation: for any unoriented skein triple $L_a,L_b,L_c\subset Y$, there are exact triangles $HF L'^-(Y,L_a)\to HF L'^-(Y,L_b)\to HF L'^-(Y,L_c)\to HF L'^-(Y,L_a)$ and the analogous triangles in the infinity and hat versions, with all arrows given by band maps. The paper proves this by establishing a local model on the torus: the mapping cone of the non-orientable band map $\tau\colon \beta_a\to\beta_b$ is morally quasi-isomorphic to $\beta_c^E$, the third attaching curve equipped with a rank-two local system $E$ determined by an oriented arc $G$. The key tool is a Heegaard Floer analogue of a recent 2-surgery exact triangle in instanton Floer homology: for a knot $K\subset Y$ and any framing $\lambda$, there is an exact triangle $HF L'^-(Y,K)\to HF^-(Y_\lambda(K))\to HF^-(Y_{\lambda+2\mu}(K))\to HF L'^-(Y,K)$, where $\mu$ is the meridian; the paper derives this as Theorem 4.5 and the main skein triangle from the same local computation. If the main theorem is right, the unreduced and reduced hat versions are the Heegaard Floer counterparts of the instanton invariants $I^\sharp$ and $I^\natural$, and iterating the triangle would produce spectral sequences from Khovanov homology to unoriented link Floer homology.

Load-bearing premise

The load-bearing premise is the local enumeration on the torus: every holomorphic triangle, quadrilateral, or pentagon not explicitly listed cancels in pairs, and the series $u=\sum_{n\ge1}U^{n^2-n}$ is invertible; if an unlisted domain contributes, $\rho$ and $\sigma$ are no longer morally inverse and the exact triangle can fail.

Editorial extensions

If this is right

  • For every unoriented skein triple, the minus, hat, and infinity versions of unoriented link Floer homology all fit into cyclic exact triangles with band-map arrows.
  • The 2-surgery exact triangle follows as a corollary, relating unoriented knot Floer homology of a knot to Heegaard Floer homology of its $\lambda$ and $\lambda+2\mu$ surgeries.
  • For planar links, the band maps agree with the deformed equivariant Khovanov band maps, so the two theories have the same local skein behavior.
  • The rank computations for unlinks, the Hopf link, and trefoils match Khovanov homology over $\mathbb{Z}/2$ in the unreduced case, consistent with the proposed Heegaard Floer analogue of the instanton invariants.
  • Exactness in the hat and infinity versions follows from exactness in the minus version by an algebraic reduction, so the skein relation is not an artifact of one flavor.

Reading between the lines

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

  • If the triangle can be iterated over a cube of resolutions, it should produce a spectral sequence from Khovanov homology to unoriented link Floer homology; the paper states this iteration as a forthcoming result, so the spectral sequence is a projected consequence rather than a theorem proved here.
  • The mod-2 setting is probably essential: the local counts are carried out over $\mathbb{F}=\mathbb{Z}/2$ and the paper leaves signs for a $\mathbb{Z}$-lift open, so the triangle may not survive integrally without additional structure.
  • The 2-surgery triangle provides a route to compare unoriented link Floer homology with ordinary Heegaard Floer homology of surgeries, which may give new constraints on surgery distances or concordance invariants, though the paper does not develop these applications.
  • The comparison with instanton invariants suggests that, over $\mathbb{Q}$, the unreduced hat version should be isomorphic to $I^\sharp$ while over $\mathbb{Z}/2$ the two theories should differ; the trefoil rank computation illustrates exactly this divergence.
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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

3 major / 4 minor

Summary. The paper defines band maps in unoriented link Floer homology HF L'^-(Y,L) over the field F = Z/2, for non-orientable, split, and merge bands between balled links, and proves an unoriented skein exact triangle in the minus, hat, and infinity versions (Theorems 1.15 and 4.2). It also derives a 2-surgery exact triangle (Theorems 1.3 and 4.5) analogous to Bhat's instanton triangle. The proof is reduced to a local torus computation, Theorem 4.3, which is established through three morally quasi-inverse pairs of twisted complexes (Claims 6.1-6.3). The local holomorphic polygon computations in Sections 7 and 8 are presented as cancellation arguments and diagrams, and Section 9 completes the proof using homological Z-gradings, an Alexander Z/2-splitting, and invariance under almost complex structures. The paper also proves that planar band maps agree with the corresponding Khovanov homology band maps.

Significance. If the local computations are correct, this is a substantial contribution to Heegaard Floer theory: it provides a Heegaard Floer analogue of the I^# skein exact triangle, gives a 2-surgery exact triangle parallel to Bhat's instanton result, and sets up a framework for potential spectral sequences from Khovanov homology to unoriented link Floer homology. The paper is carefully organized, uses external tools appropriately (the Ozsvath-Szabo triangle detection lemma, Zemke's basepoint moving maps, the HHSZ stabilization results), and gives explicit computations for unlinks, Hopf link, trefoils, and planar band maps. The main theorems are concrete and falsifiable, and the reduction to finite-dimensional torus models is a strength. However, the central geometric input consists of holomorphic polygon counts that are asserted via families and cancellations rather than fully enumerated, so the validity of the main theorem currently rests on unverified completeness of those counts.

major comments (3)
  1. [Sections 7.1-7.3, Claim 6.1 and Remark 7.1] The proof of Claim 6.1 is not a complete enumeration of the contributing holomorphic polygons. In Section 7.1 the vanishing of the relevant mu2 maps is justified by asserting that for each n the two triangles 'related by rotation by pi' cancel; in Section 7.2 the mu3 vanishings are justified by asserting that two families of quadrilaterals cancel and that the remaining quadrilaterals 'have theta as a vertex, and there are exactly two of them'; and in Section 7.3 the mu4 identities are justified by asserting that, in each case, 'there is exactly one pentagon' without any basepoints. No combinatorial Maslov-index or filtration argument is given to prove that every domain with the relevant Maslov index and vertex sequence belongs to one of the enumerated families or to a pair with equal weight. The exact triangle over FJU1,U2^{1/2}K needs these identities modulo the maximal ideal; a single unaccounted family with odd moduli count and zero basepoint multiplicity would change mu2(rho,sigma') by a non-unit and break Lemma 2.37. The invertibility of u = sum U^{n^2-n} is not an issue, since its constant term is 1; the issue is the completeness of the counts.
  2. [Section 8, Claim 6.3 and Remark 8.1] The analogous computation for Claim 6.3 is also incomplete. For mu2(e0 xi, e*1 zeta), the paper lists the contributions of the families T_n^+ and T_n^- and asserts that the families S_n^+ and S_n^- contribute zero because e*1 phi^{2k} e0 theta0 = 0, but it does not prove that these are all the triangles with the relevant vertex sequence. For mu3(e*1 zeta, theta, e0 xi'), the paper states that the two quadrilaterals highlighted in yellow 'are the only quadrilaterals that contribute' modulo U, but no enumeration is supplied that justifies this exclusivity. Since Claim 6.3 and the additional identity mu2(e0 xi, e*1 zeta) = 0 support both Theorem 4.3 and Theorem 4.5, these missing enumerations are load-bearing for the main results.
  3. [Section 9.5 and proof of Theorem 4.2] The final step of Theorem 4.3 and the exactness at Lc in Theorem 4.2 are too compressed. In Section 9.5, the maps e rho and e sigma are defined as compositions and their top homological grading components are identified, but the argument relies on unstated uniqueness claims for cycles in specified Z x Z/2 gradings, such as the assertion in Section 9.3 that Theta+ = Theta+_a + Theta+_b is the only nonzero cycle in the relevant grading. More importantly, the proof of exactness at Lc in Theorem 4.2 asserts that, for the twisted complex beta_bc := beta_b --rho--> beta_c^E, the maps tau : beta_a -> beta_bc and sigma : beta_bc -> beta_a 'are cycles and mu2(tau,sigma') and mu2(sigma,tau') are also Theta+'; this requires checking compositions such as mu2(tau,rho) and mu2(sigma,rho), which are not among the identities established in Sections 7 and 8. This step should be written out explicitly, since it is essential for exactness at the third vertex of the triangle.
minor comments (4)
  1. [Definitions 1.5, 1.15 and Theorem 4.2] The statement that the exact triangle is FJU K-linear is not fully reconciled with the chain complex definitions, which use variables U_i^{1/2} for link components and U_i for free baseballs. Since baseball types can change in a skein triple, please state explicitly the common power-series ring over which all three complexes are considered and how the identifications of the U_i variables are made.
  2. [Section 2.8.2] The sentence beginning 'Totalkabout Spinc(Y (Lsut))-summands' appears to be missing a word; it should likely read 'To talk about Spinc(Y (Lsut))-summands'.
  3. [Theorem 4.3 and Figure 4.1] The definitions of rho = e0 rho1 + e0 rho2 and sigma = e*1 sigma1 + e*1 sigma2 depend on the labels in Figure 4.1. Please ensure that the figure and its labels are legible and that the curves beta_a, beta_b, beta_c, and the intersection points rho1, rho2, sigma1, sigma2 are explicitly identified in the caption or text.
  4. [Section 3.3.5 and Remark 3.24] The condition |P cap beta| = 1 in the definition of a Heegaard triple subordinate to a merge or split band is stated without justification. A sentence explaining why this condition can always be arranged and why it is harmless would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the unoriented skein triangle is derived from local holomorphic polygon counts that are independent of the theorem being proved.

full rationale

The derivation chain runs from explicit Heegaard–Floer chain complexes and band maps to the local computation Theorem 4.3, whose proof is a collection of torus-level polygon counts (Claims 6.1–6.3, Sections 7–9), followed by the triangle detection lemma and related quasi-isomorphism arguments. The maps τ, ρ, σ, ζ, ξ, e0ξ, e1*ζ are explicit generators in fixed diagrams, not parameters fitted to force exactness in a target homology group. No equation or definition in the paper makes Theorem 1.2 true by construction: the exact triangle is concluded only after the morally-quasi-inverse compositions µ2(ρ,σ′) and µ2(σ,ρ′) are computed to be Θ+. External tools such as Ozsváth–Szabó’s triangle detection lemma, stabilization propositions from HHSZ, and basepoint-moving results from Zemke are cited as established ingredients, and none of them is used to import the unoriented skein triangle itself as a hypothesis. The author’s own forthcoming works [Naha] and [Nahb] appear only in the motivation/future-directions discussion, not in the proof of Theorem 1.2 or Theorem 4.5, so they are not load-bearing self-citations. The manuscript does contain genuine limitations and completeness concerns: Remarks 4.4 and 4.6 restrict the validity of some versions, Section 7–8 cancellations are asserted rather than fully enumerated, and Section 9.5 says the final assembly is ‘standard’. These are gaps in verification and presentation, not circularity, because the missing or abbreviated justifications do not assume the conclusion of the exact triangle; they concern the correctness of the local holomorphic counts. No fitting-to-data, definitional identification between premise and conclusion, or uniqueness theorem imported from the author’s own prior work was found.

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

No fitted parameters appear. The formal variables U_i are coefficient-ring variables, not numbers matched to data. The main proof relies on standard Heegaard Floer foundations and on two paper-specific computational premises: the local polygon counts and the absence of relevant Maslov index 0 domains. The characteristic 2 sign-free setting is explicit.

assumptions (6)
  • standard math Heegaard Floer homology and the A-infinity category of holomorphic polygon counts are well-defined and satisfy the standard A-infinity relations.
    Invoked throughout Sections 2, 6-9; analytic foundations are deferred to OS04c, Lip06, LOT16, and Seidel.
  • domain assumption Weak admissibility and positivity ensure finiteness of the sums and the well-definedness of the filtered chain complexes.
    Needed to define CF L'^- with power series coefficients and at most two nontrivial local systems; see Lemmas 2.20 and 2.23.
  • ad hoc to paper The local counts in T^2 (Claims 6.1-6.3, Sections 7-8) are correct, including the cancellation of triangle and quadrilateral families and the identity mu2(e0 xi, e*_1 zeta)=0.
    This is the load-bearing computational premise for Theorem 4.3; it is specific to this proof and not externally verified.
  • ad hoc to paper There are no relevant Maslov index 0 domains, so the cycles tau, theta, rho, sigma and canonical elements are invariant under changing the almost complex structure.
    Used in Remarks 3.12 and 3.19 and Section 9.4 to transfer computations among almost complex structures.
  • standard math The basepoint moving map is homotopic to the identity, and diffeomorphisms of D^3 fixing the boundary and an interval are isotopic rel boundary.
    Used in Lemma 3.7 and Proposition 3.8 for naturality of the chain complex under auxiliary data.
  • domain assumption All statements are over F=Z/2, where signs are ignored.
    Stated in Section 1.5 and used throughout; the author notes it is open whether the theorems hold over Z.

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Pith. "Pith review of An unoriented skein exact triangle in unoriented link Floer homology." pith.science (2026). https://pith.science/paper/XZ4XSWME

@misc{pith2026250101047,
  author       = {Pith},
  title        = {Pith review of: An unoriented skein exact triangle in unoriented link Floer homology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XZ4XSWME}},
  note         = {Machine review of arXiv:2501.01047}
}
abstract

We define band maps in unoriented link Floer homology and show that they form an unoriented skein exact triangle. These band maps are similar to the band maps in equivariant Khovanov homology given by the Lee deformation. As a key tool, we use a Heegaard Floer analogue of Bhat's recent 2-surgery exact triangle in instanton Floer homology, which may be of independent interest. Unoriented knot Floer homology corresponds to $I^{\sharp}$ of the knot in our 2-surgery exact triangle.

Figures

Figures reproduced from arXiv: 2501.01047 by the authors.

Figure 1.1
Figure 1.1. A local diagram for an unoriented skein triple Remark 1.4. We thank Fan Ye for communicating to the author that a similar analogue of I ♯ for knots in Heegaard Floer homology2 has been suggested in his Miami talk [Ye23]. 1.1. Unoriented link Floer homology. Let us define the unoriented link Floer homology groups that we study in this paper. Definition 1.5. Let L be a k-component link in a closed, oriented three-mani… view at source ↗
Figure 1.2
Figure 1.2. A local diagram for an unoriented skein triple together with the band maps Theorem 1.15. Let La, Lb, Lc ⊂ Y be an unoriented skein triple. Then there exist exact triangles in the various versions · · · → HF L′− (Y, La) → HF L′− (Y, Lb) → HF L′− (Y, Lc) → HF L′− (Y, La) → · · · · · · → HF L \′(Y, La) → HF L \′(Y, Lb) → HF L \′(Y, Lc) → HF L \′(Y, La) → · · · · · · → HF L ^′(Y, La) → HF L ^′(Y, Lb) → HF L ^′(Y, Lc) → … view at source ↗
Figure 1.3
Figure 1.3. A non-orientable band between two unknots for Manolescu’s exact triangle, and a Heegaard diagram for it 1.3. A word on the proof. We discuss some key ideas involved in the definition of the split and merge band maps and the proof of Theorem 1.15. The split and merge band maps involve links with different numbers of link components, and it is not obvious how to define them since we want exactly two basepoints on each… view at source ↗
Figures from the paper (30 more)
Figure 1.4
Figure 1.4. Figure 1.4: A non-orientable band between two unknots for our exact triangle, and a Heegaard diagram for it complex [OS08a, Proposition 6.5]). This basepoint becomes two link basepoints on the link with more components. See [PITH_FULL_IMAGE:figures/full_fig_p008_1_4.png]
Figure 1.5
Figure 1.5. Figure 1.5: We reduce the proof of Theorem 1.15 to a local computation, The￾orem 4.3. Remark 1.16. We prove Theorem 1.3 directly, by doing a model computation on the torus in the spirit of [OS04b, OS05]. We thank Ian Zemke for communicating to the author that Theorem 1.3 can als…
Figure 2.1
Figure 2.1. Figure 2.1: A local diagram for the Heegaard diagram near G Definition 2.13. We chose a basis for the nontrivial local system E. Choose the trivial basis {1} for the trivial local system. These induce R-bases on the R-Hom spaces. An R-generator (or simply a generator ) of CF −(α…
Figure 2.2
Figure 2.2. Figure 2.2: Some genus 1 examples Remark 2.16. One can define the composition maps µn without mentioning local systems, if there is at most one attaching curve with a nontrivial local system. Let us demonstrate this for µ1 (we have observed a special case in the first example of…
Figure 2.3
Figure 2.3. Figure 2.3: A local diagram for Lemmas 2.18 and 2.23 Lemma 2.18. Let α E0 0 , · · · , α Ed d be attaching curves with local systems. Consider generators fixi ∈ CF −(α Ei−1 i−1 , α Ei i ) for i = 1, · · · , d and y ∈ α0 ∩ αd. Assume that there is a domain D ∈ D(x1, · · · , xd, y)…
Figure 2.4
Figure 2.4. Figure 2.4: Standard translates of β Definition 2.33. Let β = {β 1 , · · · , βn} be an attaching curve. A standard translate β ′ = {β ′1 , · · · , β′n} of β is given by slightly translating each circle β i , such that #(β i ∩β ′j ) = 2δij . If we consider other attaching curves …
Figure 2.5
Figure 2.5. Figure 2.5: Different kinds of stabilizations that we consider. p is the point we connected sum along, and z is a basepoint. Genus stabilization is where the Heegaard surface S(Σ) is given by connected summing Σ with T 2 , and the attaching curves are S(β), which are given by th…
Figure 2.6
Figure 2.6. Figure 2.6: Some simple doubly pointed Heegaard diagrams 2.8.2. The case G ̸= ∅. Let us consider the case where G may exist. In this case, we can formally work with the Heegaard diagram (Σ, α, β, p ⊔ {∂+G}); define the corresponding L sut and s : α ∩ β → Spinc (Y (L sut)) accord…
Figure 3
Figure 3. Figure 3: is a diagram of a kind of split band that we consider. The link [PITH_FULL_IMAGE:figures/full_fig_p033_3.png]
Figure 3.1
Figure 3.1. Figure 3.1: is a diagram of a kind of split band that we consider. The link La has two link basepoints z and w, but there is also a free basepoint v in the underlying three-manifold. Band surgery along the specified band creates a new link component (the left hand side part of L…
Figure 3.2
Figure 3.2. Figure 3.2: The region H, the tangle Lf0, and the β and γ circles for Definition 3.11 and 3.17. 3.2. Non-orientable bands. Let B be a non-orientable band from a balled link L to L ′ in a three-manifold Y . Then L and L ′ have the same coefficient ring; denote it as R. We will de…
Figure 3.3
Figure 3.3. Figure 3.3: A local Heegaard diagram for non-orientable bands. Work over FJU 1/2 K and assign the weight U 1/2 to both w and z. Given a Heegaard datum with underlying Heegaard diagram (Σ, α, β, γ,u ⊔ v) subordinate to a non-orientable band B from L to L ′ , we would like to defi…
Figure 3.4
Figure 3.4. Figure 3.4: We view merge and split maps as the composition of two maps 3.3. Merge and split bands. Merge and split bands are more complicated to define. We will conceptually think of them as the composition of a map that is induced by two bands (a “mergesplit pair”), and a birt…
Figure 3.5
Figure 3.5. Figure 3.5: A schematic of a mergesplit pair B1, B2 on L 3.3.2. The map for a mergesplit pair [PITH_FULL_IMAGE:figures/full_fig_p040_3_5.png]
Figure 3.6
Figure 3.6. Figure 3.6: The region H, the tangle L0, and the β and γ circles for Definition 3.11. Define the sutured manifold obtained from Y \N(L∪B1∪B2∪v) as follows. On ∂N(L∪B1∪B2), we add “meridional” sutures for each basepoint of u, such that R− corresponds to the parts of ∂N(L∪B1∪B2) t…
Figure 3.7
Figure 3.7. Figure 3.7: A local Heegaard diagram for a mergesplit pair. Unless otherwise specified, work over FJU 1/2 1 , U1/2 2 K and assign the weight U 1/2 i to wi and zi . Heegaard datum with attaching curves β, γ can be obtained from a stabilization of [PITH_FULL_IMAGE:figures/full_fi…
Figure 3
Figure 3. Figure 3: ) [PITH_FULL_IMAGE:figures/full_fig_p043_3.png]
Figure 4.1
Figure 4.1. Figure 4.1: The genus 1 diagram for Theorem 4.3. Unless otherwise specified, work over FJU1, U1/2 2 K and assign the weight U1 to v, and U 1/2 2 to w and z. Without loss of generality, assume that Lc has one more link component than La and Lb. This is equivalent to that the two …
Figure 4
Figure 4. Figure 4: , [PITH_FULL_IMAGE:figures/full_fig_p047_4.png]
Figure 6.1
Figure 6.1. Figure 6.1: The genus 1 diagrams for Claims 6.1, 6.2, and 6.3. Unless otherwise specified, work over FJU1, U1/2 2 K and assign the weight U1 to v, and U 1/2 2 to w and z. handleslides. β0 β2 βf0 βf2 βf0 βf2 β0 β2 θ Θ + 0 Θ + 2 θe Θ + 0 Θ + 2 θe θ Proof. Let us show that the dash…
Figure 7.1
Figure 7.1. Figure 7.1: Heegaard diagrams (T 2 , βa, βb, β0, β2, β′ a , β′ b , {w, z}) and (T 2 , β0, β2, βa, βb, β′ 0 , β′ 2 , {w, z}) 7. Local computation for Claim 6.1 Consider the Heegaard diagram (T 2 , βa, βb, β0, β2, {w, z}) (together with standard translates) given by [PITH_FULL_IM…
Figure 7.2
Figure 7.2. Figure 7.2: Right triangles and their names • µ2(ξ, ζ′ ): µ3(ξ, τ, ζ′ ) = 0, µ4(θ, ξ, τ, ζ′ ) = Θ+ 0 , µ4(ξ, τ, ζ′ , θ′ ) = Θ+ 2 , µ5(θ, ξ, τ, ζ′ , θ′ ) = 0 modulo U 1/2 Remark 7.1. In fact, we can explicitly compute µ2(ζ, ξ′ ) and µ2(ξ, ζ′ ). All the µ3’s and µ5’s vanish, and t…
Figure 7.3
Figure 7.3. Figure 7.3: Left: two small triangles with weight U 1/2 ; Right: Two small quadrilaterals with weight 1 7.2. The µ3’s vanish. We will write ζ instead of ζ ′ and β0 instead of β ′ 0 , etc. since distinguishing them in this subsection is meaningless. Let us show that µ3(τ, ζ, θ), …
Figure 7.4
Figure 7.4. Figure 7.4: Relevant pentagons for µ4(τ, ζ, θ, ξ′ ) and µ4(θ, ξ, τ, ζ′ ) modulo U 1/2 . These are small perturbations of the top quadrilateral of the right hand side of [PITH_FULL_IMAGE:figures/full_fig_p055_7_4.png]
Figure 7.5
Figure 7.5. Figure 7.5: Two triangles with weight U 5/2 [PITH_FULL_IMAGE:figures/full_fig_p055_7_5.png]
Figure 7.6
Figure 7.6. Figure 7.6: Six quadrilaterals with weight U 2 . The top quadrilaterals are small perturbations of T + 2 and the bottom ones are small perturbations of T − 2 [PITH_FULL_IMAGE:figures/full_fig_p056_7_6.png]
Figure 8.1
Figure 8.1. Figure 8.1: The genus 1 Heegaard diagram (T 2 , β0, β2, β∞, β′ 0 , β′ 2 , β′ ∞, v) [PITH_FULL_IMAGE:figures/full_fig_p057_8_1.png]
Figure 8.2
Figure 8.2. Figure 8.2: The two quadrilaterals highlighted in yellow are the two quadrilat￾erals that contribute modulo U, and they add up to IdEΘ+ ∞. 8.2. The µ3’s are the identity. We compute µ3(e ∗ 1 ζ, θ, e0ξ ′ ), which is the most complicated case. If we work modulo U, then the two sma…
Figure 9.1
Figure 9.1. Figure 9.1: A cornerless two-chain D whose boundary has exactly one circle with slope 2. 9.2. An Alexander Z/2-splitting. Proposition 9.2. Any Heegaard diagram obtained from the Heegaard diagram (T 2 , βa , βb , β0 , β2 , βf0 , βf2 , β E c , {v, w, z}, G) given in Section 6 by a…
Figure 9.2
Figure 9.2. Figure 9.2: A Heegaard diagram (T 2 , βa , βb , β ′ a , β ′ b , z, w, v) 9.4. Independence under variation of the almost complex structure. We show that the statement of Theorem 4.3 is insensitive of the almost complex structure. Let us first check that the maps are invariant. I…

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Works this paper leans on

51 extracted references · 34 canonical work pages

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    Deeparaj Bhat, Surgery exact triangles in instanton theory, arXiv preprint arXiv:2311.04242 (2023)

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