Perturbed Traceless SU(2) Character Varieties of Tangle Sums

Kai Smith

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Perturbed Traceless SU(2) Character Varieties of Tangle Sums

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A cut-and-paste formula computes perturbed tangle-sum character varieties, replacing each internal circle by two intervals.

desk verdict A real gluing theorem for pillowcase Lagrangians, with an overclaimed Proposition 5.7 and a bigon count that deserves a second pass; still worth refereeing. read the letter →

arxiv 2412.06066 v1 pith:IMWFNWRX submitted 2024-12-08 math.GT

Kai Smith This is my paper · ORCID

classification math.GT MSC 57K1057K3157R58

keywords tanglesumtracelessSU(2)charactervarietypillowcaseholonomyperturbationLagrangianFloerhomologyboundingcochainsreducedsingularinstantonarborescenttangles

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

This paper establishes a cut-and-paste rule for the traceless SU(2) character varieties that appear as Lagrangians in the pillowcase. For any good pair of tangles with no corner circles, the perturbed character variety of the tangle sum $T_1+T_2$ is obtained from the unperturbed one by deleting a neighborhood of each internal circle and inserting two intervals whose pillowcase images converge to the deleted circle, with any remaining auxiliary circles shrinking to points as the perturbation tends to zero. On that basis the paper shows that arborescent tangles have piecewise-linear pillowcase images with rational slopes, and it proves that if the bounding-cochain conjecture for the pillowcase holds, then some tangle without an earring must carry a nontrivial bounding cochain. A reader should care because these Lagrangians are the proposed bridge between reduced singular instanton homology and computable Lagrangian Floer homology in the pillowcase.

What carries the argument

The central object is the tangle sum $T_1+T_2$, formed by gluing two tangles into the elementary tangle $C_3$, whose character variety fibers over the fiber product $R_{\pi_1}(T_1)\times_{[0,\pi]}R_{\pi_2}(T_2)$ over the common $\gamma$-coordinate (Theorem 3.15). The perturbation is a holonomy perturbation along a single curve $D$ in $C_3$; the mechanism that carries the argument is the zero-set $V_t=\Phi_t^{-1}(0)$ for an explicit trigonometric function $\Phi_t$, whose values near the singular locus are controlled by a sign function $s$ on the set $S$ of binary-dihedral representations. The sign of $s$ decides how each internal circle reconnects: neighborhoods of the singular pair are replaced by two intervals joining the $A^+$ and $A^-$ endpoints, while the $s=0$ locus keeps the cone-on-four-points structure (Theorem 4.13 and Corollary 4.15).

What would settle it

For a concrete good pair, such as $Q_{1/2}+Q_{-1/3}$ with perturbation $D_t$, explicitly enumerate all auxiliary components and check whether any auxiliary circle persists, intersects the other Lagrangian under every sufficiently small perturbation, or fails to shrink to a point as $t\to0$; finding one such circle would disprove Proposition 5.7 and undo the Floer-rank conclusion in Theorem 5.9.

Watch

Extended reading notes

Core claim

The paper's load-bearing claim is Theorem 4.22: for a good pair of tangles with no corner circles and sufficiently small $t>0$, the perturbed character variety $R_{D_t\cup\pi_1\cup\pi_2}(T_1+T_2)$ contains a main component obtained from $R_{\pi_1\cup\pi_2}(T_1+T_2)$ by removing a neighborhood of each internal circle $C_i$ and inserting two intervals whose images in the pillowcase converge to the image of $C_i$; every other component is an auxiliary circle whose image converges to the singular set as $t\to0$. The paper then derives two consequences: arborescent tangle character varieties are linear with rational slopes (Proposition 5.8), and, conditional on the pillowcase bounding-cochain conjecture, there exists a tangle without an earring whose bounding cochain is nontrivial, shown by computing nine intersection points and two bigons for the knot $P(-2,3,5)$ and finding Floer rank five rather than the known rank seven of $I^\natural$ (Theorem 5.9).

Load-bearing premise

The argument rests on assuming that the auxiliary circle components left undetermined by Theorem 4.22 can always be made irrelevant to Lagrangian Floer homology by an arbitrarily small perturbation of the other Lagrangian (Proposition 5.7); if for some tangle pair those circles cannot be avoided, the computed Floer homology and the nontriviality of the bounding cochain could change.

Editorial extensions

If this is right

For any good pair with no corner circles, the perturbed character variety of a tangle sum has a main component obtained from the unperturbed variety by replacing each internal circle by two intervals, with all other components collapsing to points as $t\to0$ (Theorem 4.22).
Arborescent tangles have piecewise-linear pillowcase images with rational slopes and endpoints in $(\pi\mathbb{Q})^2$, so their character varieties are algorithmically computable (Proposition 5.8).
If the pillowcase bounding-cochain conjecture holds, the bounding cochain assigned to some tangle without an earring is nontrivial; the example is the decomposition of $P(-2,3,5)$ into $dQ_{-1/2}$ and $Q_{1/3}+Q_{1/5}$ (Theorem 5.9).
Under Proposition 5.7, auxiliary components do not affect Lagrangian Floer homology, so the $t\to0$ limit of the perturbed character variety can be used in place of the actual Lagrangian in computations.

Reading between the lines

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

Editorial inference: the same surgery can likely be iterated over arbitrary arborescent diagrams, turning the paper's one-step tangle-sum move into an inductive algorithm that computes pillowcase Lagrangians for all Montesinos and arborescent knots, not only the examples checked.
Editorial inference: the existence of auxiliary components whose number and placement are undetermined suggests that the pillowcase Lagrangian is not unique up to Hamiltonian isotopy; proving the bounding-cochain conjecture may require a canonical choice or a bounding-cochain correction that absorbs these components rather than only perturbing them away.
Editorial inference: the rank-five versus rank-seven mismatch for $P(-2,3,5)$ can be used as a test case: any proposed bounding cochain for $Q_{1/3}+Q_{1/5}$ must add exactly two units of rank, and a computer search over piecewise-linear cochains could identify the minimal correction.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Kai Smith's paper gives the first general cut-and-paste description of perturbed traceless SU(2) character varieties under Conway tangle sum. Theorems 3.15 and 4.22 are genuine results: they tell you how to assemble the pillowcase image of R(T1+T2) from the images of R(T1) and R(T2), with the usual caveat that the perturbed sum may have extra circle components. The long computation of the perturbed character variety of C3 (Sections 3–4) is careful and self-contained, and the paper is honest about what it does not determine: Theorem 4.22 explicitly leaves the number of auxiliary components open.

The conditional result, Theorem 5.9, is also honestly stated: if CHKK Conjecture D holds, then some tangle without an earring must have a nontrivial bounding cochain. That is a worthwhile step toward the Atiyah–Floer reconstruction program.

Soft spots. The biggest one is Proposition 5.7. The proof only arranges the other Lagrangian to miss the auxiliary components. That is enough to kill differentials when the bounding cochain is zero, because then all generators are intersection points between the two Lagrangians. But the proposition is stated for arbitrary bounding cochains b. If b has nonzero coefficients on self-intersections of the auxiliary components with the main component, then those generators remain even when the other Lagrangian misses the auxiliary components, and polygons with a vertex on b can couple to main intersection points. The restriction b† need not be a bounding cochain. So the proposition as stated is not proved. This does not sink Theorem 5.9, which uses b=0, but it does mean the advertised 'auxiliary components can be ignored' slogan is only justified in the untwisted case.

The other soft spot is the bigon count in the proof of Theorem 5.9. It is done by eye from Figure 31. The author has a program (pcase) that computes these counts for piecewise-linear curves; it would strengthen the paper to verify this HF rank computationally, or at least to spell out the bigon identification in coordinates.

Overall, the central gluing construction is solid and the paper is worth serious refereeing. The referee should ask for a corrected or narrowed Proposition 5.7 and a more checkable bigon count. I would send it out.

Referee Report

3 major / 4 minor

Summary. The paper develops a cut-and-paste description of holonomy-perturbed traceless SU(2) character varieties for tangle sums. After computing the unperturbed character variety of the elementary tangle C3, it proves a fiber-product description of R_{π1∪π2}(T1+T2) (Theorem 3.15), then studies the effect of a specific perturbation curve D on C3. The main structural result, Theorem 4.22, states that for a good pair with no corner circles, the perturbed character variety contains a 'main component' obtained from the unperturbed variety by deleting neighborhoods of each internal circle and inserting two intervals, with any remaining components being auxiliary circles that shrink to points as the perturbation parameter tends to zero. The paper then applies this structure to Lagrangian Floer homology in the pillowcase: Proposition 5.7 asserts that auxiliary components may be ignored, Proposition 5.8 claims arborescent tangles are linear with rational slopes, and Theorem 5.9 claims that, assuming Conjecture 5.2, there must exist tangles without earrings whose bounding cochains are nontrivial, via a computation for the pretzel knot P(−2,3,5).

Significance. If the main structural and Floer-theoretic claims are correct, the paper would provide a substantial new computational tool for traceless SU(2) character varieties of tangle sums and would make progress on the CHKK Conjecture D program. The manuscript has notable strengths: the computation of the perturbation function Φ_t for C3 is explicit and detailed, the sign function s in Section 4.1.4 gives a concrete local mechanism for how the perturbation reconnects components, Theorem 3.15 and Proposition 5.8 are self-contained and do not depend on the undetermined auxiliary components, and the author provides a reproducible computer program pcase for further computations. However, the bridge from Theorem 4.22 to the Floer-theoretic applications rests on Proposition 5.7, which is not justified, and the rank computation in the proof of Theorem 5.9 appears internally inconsistent. The significance of the paper is therefore currently conditional: the structural Theorem 4.22 may well be correct, but the advertised applications are not yet established.

major comments (3)

[§5.5, Proposition 5.7] The assertion that auxiliary components may be ignored in Lagrangian Floer homology is not proved. An auxiliary component B of L := R_{Dt∪π1∪π2}(T1+T2) contributes not only intersection points with a second Lagrangian L3, but also self-intersections of L, including intersections between B and the main component; these self-intersections are generators of CF(L,L). The proof only arranges a perturbation of L3 so that its image misses the limit points {p_i}, which does not control whether auxiliary and main components intersect in the pillowcase. Moreover, the restriction b† of a bounding cochain b to the main component is not shown to satisfy the Maurer-Cartan equation (Eq. 5.3.1) for the restricted Lagrangian: terms in the Maurer-Cartan equation supported on B, and holomorphic polygons with a b-vertex on B whose remaining edges lie on L, can contribute to differentials between main-component intersection points even when L3 avoids B. Therefore the equality HF((L,b),(L3,b′)) = HF((L†,b†),(L3,b′)) is unproved, and the applications in §5.5 that ignore auxiliary components rest on this gap.
[§5.5, proof of Theorem 5.9 and Figure 31] The rank computation in the proof of Theorem 5.9 is arithmetically inconsistent. The text states that CF has nine generators and that there are two bigons whose vertices are distinct. Over F2, each such bigon contributes at most one elementary differential, so the image of the boundary map has rank at most 2 and the homology has rank at least 7, not 5 as claimed. To obtain rank(HF)=5 the author would need to exhibit additional differentials or explain how two bigons reduce the dimension by four. As written, the comparison rank(I♮(P(−2,3,5)))=7 does not force b2 to be nonzero, because the unperturbed computation appears to give HF rank at least 7.
[Abstract, Introduction, and Theorem 4.22] The paper promises a method to compute the perturbed character variety of a tangle sum, but Theorem 4.22 identifies only a subspace of R_{Dt∪π1∪π2}(T1+T2): the main components. The theorem explicitly leaves the number and placement of auxiliary circle components undetermined, and Section 5.5 concedes that 'At first the result of Theorem 4.22 may seem less than helpful because it does not determine the number of auxiliary components.' Since Proposition 5.7, which is the only mechanism proposed for handling these components in Floer-theoretic computations, is unproved (see the first major comment), the full computation of the perturbed character variety and the subsequent HF computation in Theorem 5.9 are not established. The structural theorem may be correct, but it does not yet deliver the complete cut-and-paste computation announced in the abstract.

minor comments (4)

[Proposition 3.12, second and third bullets] The image formulas state p3(ρ) = (γ(ρ1), γ(ρ1)+γ(ρ2)), but the proof and the analogous statement in Theorem 3.15 use the θ-coordinates. These bullets should read (γ(ρ1), θ(ρ1)+θ(ρ2)).
[Lemma 3.25] The condition for the existence of a corner circle is written as p(W^1_∅) ∩ W^2_∅ ≠ ∅; since W^2_∅ is a subset of the character variety rather than of the pillowcase, the intended statement is p(W^1_∅) ∩ p(W^2_∅) ≠ ∅.
[Lemma 4.18] The proof refers to 'the proof of Theorem 3.8', but the relevant statement is Lemma 3.8, not Theorem 3.8.
[Section 5.5, proof of Theorem 5.9] The notation 'rank(HF^1(K))' appears to be a typo; the surrounding text concerns HF((R♮(T1),0),(RD(T2),0)), not a separate invariant HF^1(k).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main constructions are explicit cut-and-paste computations, and the only conditional step is stated as conditional on an external conjecture.

full rationale

The paper's central derivation chain is self-contained. The unperturbed tangle-sum statement (Theorem 3.15) is obtained by explicit computation of the fiber product of character varieties of the tangles and of C3, using explicit gluing parameter formulas from Lemma 2.32 and explicit spherical-trigonometry computations in Lemmas 3.7 and 3.8. The perturbed statement (Theorem 4.22) is derived from an explicit trace formula (Equation 4.1.5) for the perturbation curve D and a local analysis of the zero set Vt, including explicit Hessian computations at corners (Theorem 4.16). No parameter is fitted to a target answer, and no component of the construction is normalized so as to force the claimed pillowcase images or slopes. The linearity of arborescent tangles (Proposition 5.8) follows from explicit slope arithmetic for rational tangles (Proposition 2.27, Lemmas 2.28, 2.29, 2.30) and the additive formulas in Theorem 3.15, rather than from assuming the conclusion. The undetermined auxiliary components of Theorem 4.22 are explicitly acknowledged in Section 5.5 ('At first the result of Theorem 4.22 may seem less than helpful because it does not determine the number of auxiliary components'), and Proposition 5.7 offers a perturbation argument for ignoring them in Floer-theoretic computations; whether that argument is fully justified is a correctness question, not a circularity. Theorem 5.9 is explicitly conditional on Conjecture 5.2, which is an external conjecture of Cazassus, Herald, Kirk, and Kotelskiy, and it derives nontriviality of bounding cochains from explicit Lagrangian intersection and bigon counts, so the conjecture is not smuggled in as a conclusion. The only self-citation is the author's software pcase [Smi22], used as a computational tool for examples; it is not load-bearing for any theorem proof. The paper is therefore not circular.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; the construction is analytic and parameter-free. The paper relies on established gauge-theoretic regularity and gluing results as black boxes, and the main application theorem is explicitly conditional on CHKK's Conjecture D. No new physical or mathematical entities are introduced.

assumptions (4)

domain assumption Theorem 2.35 (Herald-Kirk): for a 2-tangle in a homology ball, arbitrarily small holonomy perturbations make the (Z/2,Z/2) stratum a Lagrangian immersion into P* and empty the (Z/2,U(1)) stratum.
Imported from [HK18, Cor D] and used as the basis for existence of good pairs in Proposition 3.17 and throughout Sections 4 and 5. It is a nontrivial gauge-theoretic regularity theorem, not proved in this paper.
standard math Gluing character varieties via double coset stabilizers (Lemma 2.32 and Lemma 2.34).
From [CHK22, Lemma 6.1], used to identify fibers in Theorems 3.15 and 3.19 and to justify homeomorphism statements for tangle sums.
domain assumption [CHK22, Theorem A]: the relevant relative character varieties map into products of pillowcases as Lagrangian immersions.
Used in Corollary 4.11 and Proposition 3.20 to conclude immersedness of M and of perturbed models in the pillowcase.
domain assumption Conjecture 5.2 (CHKK20, Conjecture D).
The statement of Theorem 5.9 is explicitly conditional on this conjecture. It is not used for the character variety computations but is a load-bearing premise for the application to bounding cochains.

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Pith. "Pith review of Perturbed Traceless SU(2) Character Varieties of Tangle Sums." pith.science (2026). https://pith.science/paper/IMWFNWRX

@misc{pith2026241206066,
  author       = {Pith},
  title        = {Pith review of: Perturbed Traceless SU(2) Character Varieties of Tangle Sums},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IMWFNWRX}},
  note         = {Machine review of arXiv:2412.06066}
}

abstract

If a link $L$ can be decomposed into the union of two tangles $T\cup_{S^2} S$ along a 2-sphere intersecting $L$ in 4 points, then the intersections of perturbed traceless SU(2) character varieties of tangles in a space called the pillowcase form a set of generators for Kronheimer and Mrowka's reduced singular instanton homology, $I^\natural$. It is conjectured by Cazassus, Herald, Kirk, and Kotelskiy that with the addition of bounding cochains, the differential of $I^\natural$ can be recovered from these Lagrangians as well. This article gives a method to compute the perturbed character variety for a large class of tangles using cut-and-paste methods. In particular, given two tangles, $T$ and $S$, Conway defines the tangle sum $T+S$. Given the character varieties of $T$ and $S$, we show how to construct the perturbed character variety of $T+S$. This is done by first studying the perturbed character variety of a certain tangle $C_3$ properly embedded in $S^3$ with 3 balls removed. Using these results, we prove a nontriviality result for the bounding cochains in the conjecture of Cazassus, Herald, Kirk, and Kotelskiy.

Figures

Figures reproduced from arXiv: 2412.06066 by the authors.

**Figure 2.** A diagram for the tangle C3. On the right is C3 along with curve D which defines a perturbation of R(C3). Even if the perturbations π1 and π2 are nontrivial, in almost all cases, the resulting character variety Rπ1∪π2 (T + S) is not a manifold. The next goal is to apply suitable perturbations so that the perturbed character variety is manifold and maps into the pillowcase as a Lagrangian. The first step in this pr… view at source ↗

**Figure 3.** Theorem 3.26 tells us that a perturbed character variety Rπ(T + S) is a 1-manifold except for a collection of subspaces which map to the pillowcase to look like (A). Theorem 4.22 says that by applying an additional perturbation π ′ , the character variety Rπ+π′(T + S) can be computed up to regular homotopy by replacing the regions that look like (A) by regions that look like (B). perturbation goes to zero, each must… view at source ↗

Figures from the paper (23 more)

**Figure 4.** Figure 4: A decomposition of P(−2, 3, 5) and the corresponding immersed Lagrangians of pillowcase. Acknowledgments. The author would like to thank Paul Kirk for suggesting the initial problem which grew into this article and for the countless hours of helpful discussion. Additi…

**Figure 5.** Figure 5: A slightly distorted picture of S and the loops that generate its fundamental group. 2.4. The Pillowcase. A particularly important example is the traceless character variety of S ∼= (S 2 , 4). Abusing notation, let a, b, c, and d denote the meridians around the punctur…

**Figure 6.** Figure 6: The pillowcase and its parametrization. pillowcase and will be denoted by P c . P c is the 0-dimensional stratum of the pillowcase. Let P ∗ denote the smooth 2-dimensional stratum of the pillowcase, P \ P c . P ∗ has a symplectic structure inherited from the standard s…

**Figure 7.** Figure 7: The perturbation curves which induce the shearing perturbations. • {D′ i } is a family of pairwise disjoint embedding S 1 × D2 → X \ T; • fi is a smooth, odd, 2π-periodic function; • ti ∈ R. The resulting perturbed character variety will only depend on the embeddings D…

**Figure 8.** Figure 8: Rational tangle of slope 0, Q0. Thus it follows that ρ(a ′ ) = ρ(pap) = e 2tf(−γ)k i ρ(b ′ ) = ρ(pbp) = e (2tf(−γ)+γ)k i. Then by Equations 2.4.3 and 2.4.4, γ ′ = ∠ρ(a ′ )ρ(b ′ ) = γ (2.5.3) θ ′ = ∠ρ(b ′)ρ(a ′ )ρ(c ′ ) = θ − 2tf(γ). Suppose (X, T) is a tangle such that…

**Figure 9.** Figure 9: The tangles T1 and Tn as described in Lemma 2.28 [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

**Figure 11.** Figure 11: (A) shows a certain tangle cobordism. The tangle (B) can be obtained by gluing tangle (C) to tangle (A). Proof. a ′ = c, b ′ = a, d ′ = b, and c ′ = d. Conjugating gives a ′ = i, b ′ = e −θk i, and c ′ = e γk i. So θ ′ = γ and γ ′ = −θ. Thus, s ′ = − 1 s . □ For a tan…

**Figure 13.** Figure 13: The tangle C3 In the literature of arborescent knots, C3 is also known as the (hollow) elementary tangle [Thi91, BS16]. The tangle sum, T1 + T2, is constructed by gluing tangles T1 and T2, into C3 along S1 and S2. Therefore, a representation ρ ∈ R(T1 + T2) can be deco…

**Figure 14.** Figure 14: The space V. The spherical coordinate is given by Γ(α, β) and the radial coordinate is given by γ. The extra S 1 coordinate comes from θ. The boundary circles of the shaded annulus are identified, giving a torus. Thus, {Γ( e γ, θ, α, β) ∈ P | cos α = 0} = {(i, b, c, x…

**Figure 15.** Figure 15: Spherical triangle formed by the images of a, c, and x. (3) This follows the same logic as the last case, but if z3 ∈ {0, π}, then z2 + z3 = z2 − z3 in S 1 and so ρ1 = ρ2. On the other hand, if z2 ∈ {0, π}, −iΓ( e z1, z2 + z3, π 2 , z2))i = Γ( e −z1, −z2 − z3, − π 2 ,…

**Figure 19.** Figure 19: The gray box represents N γ0 ϵ and the red represents V. The intersection of red lines in the middle of the square gives S γ0 . Figures 17a and 17b show representations ρ1 ∈ R(T) and ρ2 ∈ R(S) with the same γcoordinate. Figure 17c shows the unique representation [ρ1 …

**Figure 20.** Figure 20: C3 with perturbation curve D in red. 4.1.1. Fundamental Group Calculation. The fundamental group of C3 with the perturbation curve D is calculated using the Wirtinger presentation of the diagram in [PITH_FULL_IMAGE:figures/full_fig_p036_20.png]

**Figure 21.** Figure 21: For x ∈ S, these are three possible ways that Ψ−1 ϵ (x) can intersect Vt . and α ± t (θ, β) ∈ [ π 2 − ϵ, π 2 + ϵ]. For perturbation t, we want to know whether α + t or α − t is bigger. To do this, we calculate d dtα ± t |t=0. With θ, β, and γ0 fixed, consider Φ as a …

**Figure 22.** Figure 22: The torus S γ0 , color coded by the sign of s. The blue regions have s > 0, the red regions have s < 0, the black lines have s = 0. The green points denote the points in V 0 (and so s = 0 at those points). • If s(x) = 0, the intersection is a cone on four points, like…

**Figure 23.** Figure 23: One step of the process detailed in Theorem 4.22. for the abelian points) of X are of the form p ρ i j A± . Let I i 1 be an interval whose endpoints are identified with p ρ i 1 A+ and p ρ i 2 A− . Similarly, let I i 2 be an interval whose endpoints are identified with…

**Figure 24.** Figure 24: (A) shows the application of Theorem 4.22 to R(Q1 2 + Q− 1 3 ) (whose unperturbed character variety is shown in Figure 16c). The character variety splits into two components whose images in the pillowcase are shown in (B) and (C). (a) (b) [PITH_FULL_IMAGE:figures/ful…

**Figure 25.** Figure 25: The limits of the components of Rt(Q1 2 + Q− 1 3 ) as t → 0. Note that here the red components are images of intervals in the character variety, compared to the unperturbed case in Figure 16c where they were the images of circles. Remark. The perturbed character varie…

**Figure 26.** Figure 26: The perturbation curves D, D′ , and D′′ respectively [PITH_FULL_IMAGE:figures/full_fig_p055_26.png]

**Figure 27.** Figure 27: The earring cobordism, ♮. shown in [PITH_FULL_IMAGE:figures/full_fig_p055_27.png]

**Figure 28.** Figure 28: The standard bigon, B. by setting the grading of x to σ(K) mod 4 and that by doing so CH♮ ∼= CI♮ as Z/4 graded groups. So rank(I ♮ (K)) ≤ rank(CF(L1, L2)) in each grading. Let gr(y) denote this grading of y ∈ I. Let B be the standard bigon shown in [PITH_FULL_IMAGE:f…

**Figure 29.** Figure 29: (A) shows a decomposition of the unlink (S 3 , U2) into two copies of Q0. (B) shows the maps of Rπ(Q0) and R♮ (Q0) into the pillowcase where π is a shearing map. (C) shows the differential that appears when allowing for the bounding cochain b. (a) (b) [PITH_FULL_IMAG…

**Figure 30.** Figure 30: The numerator and denominator closures of a tangle. For all examples computed in [HHK14, HHK18, CHKK20], if (S 3 , L) = (D3 , T1) ∪(S2,4) (D3 , T2) where (D3 , T2) is a rational tangle, HF((Rπ(T1), 0),(R ♮ π(T2), 0)) is isomorphic to the known or conjectured value of …

**Figure 31.** Figure 31: This depicts the calculation of the Lagrangian Floer homology associated to a particular decomposition of P(−2, 3, 5). The Lagrangian intersections of R ♮ π(Qd− 1 2 ) (red) and Rt(Q1 3 + Q1 5 ) (blue). Both figures show the same Lagrangians, but highlight a different…

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Forward citations

Cited by 2 Pith papers

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The instanton homology of the $(-2,3,q)$ pretzel knots and computed bounding cochains in the pillowcase
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For every odd q≥3, the reduced singular instanton knot homology of P(-2,3,q) has rank q+2, and explicit pillowcase bounding cochains are computed that cancel or create one differential to match this rank.
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Pith tools

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

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