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Linking disks, spinning vortices and exponential networks of augmentation curves

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Pith's one-line read The quiver of open topological strings can be computed from the exponential-network data of the augmentation curve: vertices are stabilized kinky vortex paths, fugacities are their central charges, and the adjacency matrix is their…

desk verdict A serious proposal with a real circularity: the central-charge map is solid, but the intersection-pairing conventions are fitted to known quivers, so the C_ij agreement is not independent evidence. read the letter →

arxiv 2412.14901 v1 pith:7O2V5XA3 submitted 2024-12-19 hep-th math.SG

Kunal Gupta , Pietro Longhi This is my paper · ORCID
classification hep-thmath.SG MSC 81T3014N35 PACS 11.25.-w
keywords opentopologicalstringsaugmentationcurvesexponentialnetworkskinkyvorticesquiveradjacencymatrixknotsandquiversLMOVinvariants3d-3dcorrespondence
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 proposes a mirror derivation of the quiver description of open topological strings from the geometry of the augmentation curve. Its central claim is Conjecture 1: whenever a curve admits a quiver presentation, the quiver vertices correspond one-to-one with a stabilized basis of (ii,1) kinky vortex paths on the curve, each fugacity is $c_j = \frac{1}{x_{\mathrm{theory}}}\exp\!\left(\frac{Z_{a_j}}{2\pi i}\right)$, and the adjacency matrix equals the intersection matrix of the paths. Because exponential networks compute all of this data from the defining polynomial of the curve, the proposal is a constructive algorithm that works without any input from the known answer. The authors verify the conjecture on framed toric branes in $\mathbb{C}^3$ and the resolved conifold and on framed conormal Lagrangians of the trefoil and figure-eight knots, recovering the known quiver in every case.

What carries the argument

The machinery is the dictionary of Conjecture 1 together with the computational tool that produces its ingredients. An augmentation curve $\Sigma$ is the algebraic curve that describes the moduli space of an A-brane on the Lagrangian; its exponential network is a web of trajectories on the base $x$-plane that encodes the BPS spectrum of kinky vortices. Kinky vortices are BPS solitons on a cylinder that interpolate between two vacua and carry quantized flux; the (ii,1) sector consists of those with unit flux in a fixed vacuum. The algorithm stabilizes the spectrum near $x_{\mathrm{theory}}\to 0$, selects the basis paths $a_j$ with CFIV index (the integer-valued BPS counting index) of unit norm, computes fugacities from central charges, and defines $C_{ij}$ as the writhe difference $\langle a_i,a_j\rangle = \mathrm{wr}(a_i a_j)-\mathrm{wr}(a_i)-\mathrm{wr}(a_j)$ of resolved concatenated paths.

What would settle it

If the intersection matrix is computed for a quiver-presentable curve where two (ii,1) solitons have exactly equal phases at the stabilized theory point, and the two allowed concatenation orderings give different matrices, then the pairing is convention-dependent and Conjecture 1 would be false as stated; this check can be done directly on the exponential networks of the trefoil or figure-eight curves at a framing not discussed in the paper.

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Extended reading notes

Core claim

Conjecture 1 states that for an augmentation curve $\Sigma=\{(x,y)\in(\mathbb{C}^*)^2: A(x,y,u)=0\}$ that admits a quiver presentation, the quiver is encoded in the exponential network of the curve. The vertices of the quiver are in one-to-one correspondence with a distinguished stabilized basis of (ii,1) kinky vortex paths $a_j$ on the logarithmic cover $\widetilde{\Sigma}$; the quiver fugacities are $c_j = \frac{1}{x_{\mathrm{theory}}}\exp\!\left(\frac{Z_{a_j}}{2\pi i}\right)$, where $Z_a=\frac{1}{2\pi R}\int_a \log y\, d\log x$; and the adjacency matrix is the intersection matrix $C_{ij} = \langle a_i,a_j\rangle$ computed by resolving, shifting, and concatenating the projected soliton paths. The authors interpret this as the mirror of the open-string statement that holomorphic disks with Lagrangian boundary generate the LMOV spectrum, with boundary linking given by the quiver adjacency matrix. They test the conjecture by explicit exponential-network computations for framed toric branes in $\mathbb{C}^3$ and the resolved conifold and for framed conormal Lagrangians of the trefoil and figure-eight knots, including Q-deformed curves, and recover the known quivers in every case.

Load-bearing premise

The load-bearing premise is that the counterclockwise concatenation rule and the writhe sign convention are the physically correct choices for defining intersections of paths, rather than choices tuned to reproduce the known quiver.

Editorial extensions

If this is right

  • The exponential network of an augmentation curve determines the quiver data of the open topological string: vertex count, fugacities, and adjacency matrix, with no additional input.
  • The linking matrix of basic holomorphic disks on the Lagrangian equals an intersection matrix of calibrated 1-chains on the curve, so the quiver becomes a geometric invariant of the augmentation curve once the resolution and concatenation conventions are fixed.
  • The same algorithm applies to Q-deformed augmentation curves, so quivers can be computed in the deformed family and then specialized to the physical augmentation curve.
  • For toric branes and the trefoil and figure-eight conormals, the recovered quivers agree with the known ones, supporting the identification of open-string disk instantons with the stabilized (ii,1) sector of kinky vortices.

Reading between the lines

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

  • Editorial: Applied to augmentation curves whose quiver is not yet known, the algorithm would be a prediction machine; conormal Lagrangians for more complicated knots are the natural next test cases.
  • Editorial: The counterclockwise concatenation rule carries most of the conceptual weight; turning it into a statement forced by the geometry of the logarithmic cover would elevate Conjecture 1 to a theorem.
  • Editorial: Curves with higher-degree polynomial branch points have richer trajectory webs, so extending the worked examples there would stress the resolution and intersection rules beyond the current evidence.
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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 / 6 minor

Summary. The paper proposes a mirror-theoretic derivation of the knots-quivers correspondence from exponential networks of augmentation curves. Conjecture 1 identifies the vertices of the quiver with a stabilized basis of (ii,1) kinky-vortex soliton paths a_j on the log-covering of the augmentation curve, identifies quiver fugacities via c_j = (1/x_theory) exp(Z_{a_j}/2πi) in Eq. (3.7), and identifies the quiver adjacency matrix with the intersection matrix C_ij = <a_i, a_j>. Section 3.2 gives a four-step algorithm for extracting vertices, fugacities, and intersections from exponential-network data, and Section 4 tests the conjecture on framed toric branes in C3 and the resolved conifold and on framed trefoil and figure-eight conormal branes, reporting perfect agreement with known quivers from the knots-quivers literature. The paper also develops appendices on Chern-Simons vortex angular momentum, resolution of projected paths, CFIV cancellations, and off-diagonal intersection computations.

Significance. If Conjecture 1 were established, it would give a genuinely new geometric handle on the quiver/linking structure of open topological strings, connecting M2-brane disks to calibrated 1-chains on augmentation curves and providing a constructive mirror-side algorithm for the knots-quivers correspondence. The paper has real strengths: the central-charge/fugacity part of the conjecture is computed directly from exponential-network integrals without tuning and matches the known specializations; the examples are worked in substantial detail; and the appendix material on branch-point resolutions and CFIV cancellations is useful. However, the intersection-pairing claim, which carries the quiver structure, is not yet independently established because the pairing conventions in Section 3.4 are calibrated against the very examples later used as evidence. The paper is best read as a strong and detailed computational proposal rather than as a derivation of the quiver from the curve alone.

major comments (3)
  1. [Section 3.4.2 and Eq. (3.25)] The pairing <a_i,a_j> is not shown to be an invariant of the augmentation curve alone. The choice of counterclockwise concatenation is justified by the statement that it is 'the correct choice for recovering the quiver' (Section 3.4.2, before Eq. (3.24)), and the writhe sign convention in Eq. (3.25) is said to be 'adapted to recover quiver adjacency matrices in examples discussed below.' Because the same examples (C3, resolved conifold, trefoil, figure-eight) are subsequently used as tests of Conjecture 1(3), the agreement of the computed C_ij is not an independent confirmation. The authors should either derive the concatenation and writhe-sign conventions from an a priori geometric or physical principle, or freeze the conventions and test them on a quiver that was not used in their calibration.
  2. [Remark 1, Section 3.3.3] The uniqueness argument for the basis paths a_k relies on deforming the Q-deformed curve from c_j -> 0 back to the original values and on tracking soliton-antisoliton pairs by CFIV invariance. However, the c_j are precisely the quiver fugacities that the algorithm is supposed to output, and the Q-deformation data (2.20) include the adjacency matrix and the fugacities. Using these data to select the unique path for each vertex makes Conjecture 1(1) depend on the very quiver data it claims to derive. The paper needs a definition of the stabilized basis that does not presuppose the quiver, or an explicit statement of which quiver data are treated as input at each stage of the algorithm.
  3. [Section 4.4] The figure-eight test is performed on the Q-deformed curve (4.30) with deformation parameters (4.32), not on the original augmentation curve (4.28); the text states that the original curve is degenerate and that both factors must be kept. This is a reasonable consistency check only if the deformation data are known independently, but in the present framework the deformation data are part of the output of Conjecture 1. The section should clarify what is assumed as input in this example and how the deformation parameters c_2, c_5 and the theory point are fixed without already knowing the quiver.
minor comments (6)
  1. [Section 3.3.3, Conjecture 1(1)] 'CIFV index' should read 'CFIV index.'
  2. [Section 3.4.3, after Eq. (3.25)] 'where the write of each path' should read 'where the writhe of each path.'
  3. [Section 4.2.3] The paragraph begins 'In framing f = 0' but the example is framing -1; this appears to be a typo.
  4. [Section 4.4.1] The sentence introducing the figure-eight curve says 'for the conormal brane of the trefoil knot' but the section concerns the figure-eight knot.
  5. [Section 2.2] The reference 'Figure fig:multi-cover-skein' is undefined; the figure label should be supplied.
  6. [Section 4.1.3] Ellipses appear in place of explicit central-charge expressions; these formulas should be filled in.

Circularity Check

2 steps flagged · score 6.0 of 10

Section 3.4's intersection-pairing conventions are chosen to reproduce known quiver matrices, and the same known quivers are then cited as confirming the adjacency-matrix part of Conjecture 1.

  1. fitted input called prediction [Section 3.4.2 (eq. 3.24) and Section 3.4.3 (eq. 3.25)]
    "As it turns out, the correct choice for recovering the quiver is counterclockwise concatenation, which according to above also imposes the decreasing phase ordering of concatenation among projected paths. Therefore we define the concatenation of two paths of type (ii, 1) as follows (oriented from left to right) ajak if ϑaj < ϑak . ... This convention is adapted to recover quiver adjacency matrices in examples discussed below."

    The intersection pairing ⟨a_i,a_j⟩ is not a uniquely defined geometric invariant before choosing a resolution, capping, and concatenation ordering, as Section 3.4 itself states. The authors fix the ordering and the writhe sign convention by explicitly requiring that they reproduce quiver adjacency matrices. The same known matrices from the knots-quivers literature are then used as the check in Section 4. Thus the computed C_ij is forced, at least in part, by conventions calibrated to the expected output, and the reported 'perfect agreement' is not independent evidence for Conjecture 1(3). The vertex and fugacity parts of the conjecture are less affected, since central charges are computed without this tuning.

  2. self definitional [Remark 1, Section 3.3.3]
    "Suppose that Σ admits a quiver description ΣQ. Then it is possible to write it as an algebraic curve A(x, y, c1, . . . , cm) = 0 and by taking cj → 0 except for a distinguished value of j, turns the curve into 1 − y − cj(−y)Cjj x = 0 ... Next, we can turn back on the other cj̸=k ̸= 0 up to the original values that determine Σ as a specialization of ΣQ."

    The algorithm needs a unique calibrated path a_k for each quiver vertex, but the selection of that path is made by deforming the Q-deformed curve whose parameters c_j are exactly the quiver fugacities that Conjecture 1(2) is supposed to output, and whose limit curve contains the self-linking C_jj. In the examples, these c_j and the matrix C are taken from the known quiver description. The choice of representative affects the concatenations and hence the computed intersection matrix, so part of the advertised derivation of C_ij presupposes the data it claims to derive.

full rationale

The paper does substantial independent work: the vertices are identified with stabilized (ii,1) exponential-network solitons, and the fugacities c_j are computed from central-charge integrals (3.7)-(3.9) without fitting. If only those parts were at issue, the circularity score would be low. However, the adjacency-matrix claim C_ij = ⟨a_i,a_j⟩ is the load-bearing part that converts network data into the quiver, and it is not a uniquely defined geometric pairing until the conventions of Section 3.4 are fixed. Section 3.4.2 chooses counterclockwise concatenation by declaring it 'the correct choice for recovering the quiver', and Section 3.4.3 adopts a sign rule 'adapted to recover quiver adjacency matrices in examples discussed below'. The same known quiver matrices from [6,7] are then used as the benchmarks in Section 4. Moreover, Remark 1 uses the Q-deformed curve, with fugacities c_j that are themselves outputs of the algorithm, to select the unique representative paths. Hence the agreement of the computed C_ij is not an independent confirmation of the intersection-matrix part of Conjecture 1; it is partially circular. A new prediction for a quiver not already used to tune the conventions would be needed to remove this circularity. The self-citation of [16] for the stabilization mechanism is not separately penalized: it is prior work rather than a definitional reduction of the present claim.

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

No numerical parameters are fitted to the quiver data; the moduli Q, x_theory, and the epsilon deformations are generic choices used to place the theory in a stable regime. The load-bearing input is a set of background results from previous work on 3d-3d duality, exponential networks, and knots-quivers correspondence, plus hand-chosen conventions for path resolution and concatenation. The latter are the main reason the circularity burden score is non-zero.

assumptions (5)
  • domain assumption The 3d-3d correspondence maps M5 branes wrapping L to a 3d N=2 QFT T[L], identifying open topological string Hilbert spaces with vortex Hilbert spaces.
    Used in Section 2.1 (eq. 2.1); this background framework is cited from [12-15].
  • domain assumption Exponential networks compute the full BPS spectrum of kinky vortices through the nonabelianization map A(x,y,u) to {Z_a, mu_a}.
    Invoked in Section 2.4 (eq. 2.30), relying on [10,11].
  • domain assumption In the limit x to 0, the (ii,n) sector of kinky vortices reproduces standard BPS vortices.
    Section 2.1 (eq. 2.7); this is the authors' earlier result [16] and is essential for connecting quiver vertices to basic disks.
  • ad hoc to paper For a quiver-presentable augmentation curve, each quiver vertex corresponds to a unique (ii,1) soliton path with |mu| = 1 after cancellations.
    Remark 1 in Section 3.3.3 provides a deformation argument, but the uniqueness of the calibrated path in the presence of boson-fermion cancellations is assumed.
  • ad hoc to paper The intersection pairing <a_i,a_j> is well-defined via the chosen resolution, capping paths, and counterclockwise concatenation, and coincides with the quiver adjacency matrix.
    Sections 3.4.2 and 3.4.3 fix these choices explicitly to recover known quivers; invariance is checked in examples rather than proven in full generality.

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Pith. "Pith review of Linking disks, spinning vortices and exponential networks of augmentation curves." pith.science (2026). https://pith.science/paper/7O2V5XA3

@misc{pith2026241214901,
  author       = {Pith},
  title        = {Pith review of: Linking disks, spinning vortices and exponential networks of augmentation curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7O2V5XA3}},
  note         = {Machine review of arXiv:2412.14901}
}
read the original abstract

We propose a mirror derivation of the quiver description of open topological strings known as the knots-quivers correspondence, based on enumerative invariants of augmentation curves encoded by exponential networks. Quivers are obtained by studying M2 branes wrapping holomorphic disks with Lagrangian boundary conditions on an M5 brane, through their identification with a distinguished sector of BPS kinky vortices in the 3d-3d dual QFT. Our proposal suggests that holomorphic disks with Lagrangian boundary conditions are mirror to calibrated 1-chains on the associated augmentation curve, whose intersections encode the linking of boundaries.

Figures

Figures reproduced from arXiv: 2412.14901 by the authors.

Figure 1
Figure 1. Holomorphic disks Di with linking boundaries, and corresponding quiver. of calibrated 1-chains, whose detailed definition can be found in Section 3. We test our conjecture in sev￾eral examples, including framed toric branes in C 3 and the resolved conifold, and framed knot conormal Lagrangians of the trefoil and figure-eight knots. In all cases, we find a perfect match with the quiver description of the open topolog… view at source ↗
Figure 3
Figure 3. The multi-cover skein relation. The reorganization of LMOV invariants to a finite set of generators (2.10) is known to hold for spe￾cific types of Lagrangians and Calabi-Yau threefolds studied in earlier literature [3, 34–36], such as for toric branes in Calabi Yau threefolds with b4 = 0 [37], for certain knots and link conormal Lagrangian in the resolved conifold [6], and for certain knot complements viewed as zero… view at source ↗
Figure 4
Figure 4. The theory T[Q] corresponding to a quiver. Each sector carries a global ‘topological’ symmetry U(1)top, with associated flavour fugacities denoted by (x1, . . . , xm) corresponding to exponentiated Fayet-Ilioupoulos couplings. A BPS vortex carries a quan￾tized magnetic flux (i.e. U(1)m top charge) encoded by non-negative integers (d1, . . . , dm) ∈ N m. The cor￾responding grading of Hvortex is reflected by the quive… view at source ↗
Figures from the paper (33 more)
Figure 5
Figure 5. Figure 5: A Q-deformation of T[L]. In particular, vortex partition functions coincide upon a suitable identification of fugacities P ( ) (x1, x2, q) = P ( ) (x1, x2, q−1x1x2, q). (2.19) This relation can be viewed from several angles • Mathematically this can be understood as a …
Figure 6
Figure 6. Figure 6: The family of Q-deformations of the augmentation variety. We will view Σ as a finite-degree ramified covering of C ∗ with base coordinate x. A choice of (x, Q) fixes a theory T[L], and sheets yi(x) correspond to a discrete set of massive vacua for the theory.12 For a f…
Figure 7
Figure 7. Figure 7: A schematic picture of exponential networks. [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: Relations between augmentation varieties and quivers. In blue: the standard route [ [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: Left: Two solitons a, b of type (ii, 1). To compute mutual intersections (red dot) we consider the concatenation obtained by shifting b by one unit in the logarithmic label of its endpoints, as explained in Sections 3.3-3.4. The self-intersection of a denoted by a blue…
Figure 10
Figure 10. Figure 10: Walls of marginal stability, and stabilised and unstabilised theory points for [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: An (ii, 1) soliton web from the exponential network for the mirror curve of the resolved conifold, see Section 4.2. The orange cross is a branch point, the dashed orange line is a polynomial branch cut, the black dot is the puncture, the black dashed line is a logarit…
Figure 12
Figure 12. Figure 12: Zoomed in near xtheory, the translucent green and pink trajectories belong two different (ii, 1) soliton webs, with their base logarithmic indices chosen so that concatenation of soliton paths is possible. As mentioned in the text, the counterclockwise concatenation i…
Figure 13
Figure 13. Figure 13: Intersection numbers according to the right hand rule [PITH_FULL_IMAGE:figures/full_fig_p022_13.png]
Figure 14
Figure 14. Figure 14: The (ii, 1) trajectory of the exponential network of C 3 in framing f = 0. In red: the resolved projections of the soliton path a1 and its shifted copy a [+1] 1 , concatenated with each other. A subtle but crucial detail is that the projected paths must circle counter…
Figure 15
Figure 15. Figure 15: The (ii, 1) trajectory of the exponential network of C 3 in framing f = 1. In red: the resolved projections of the soliton path a1 and its shifted copy a [+1] 1 , concatenated with each other. -j ij i,0 i,1 i,1 i,2 j,0 j,1 j,1 j,1 j,2 j,0 +1 +1 [PITH_FULL_IMAGE:figur…
Figure 16
Figure 16. Figure 16: The exponential networks of C 3 in framing f = 2. Only trajectories that belong to the soliton web aW 1 are shown. In red and blue: the resolved projections of the soliton path a1 and its shifted copy a [+1] 1 , concatenated with each other. 4.1.4 Framing 2 In framing…
Figure 17
Figure 17. Figure 17: Soliton webs aW i for the resolved conifold in framing f = 0 shown as grey network trajectories. Resolved projections of soliton paths, and the concatenation with their own their logarithmic shifts are also shown. The black dot denotes the puncture at x = 0. 4.2.3 Fra…
Figure 18
Figure 18. Figure 18: Soliton webs aW i for the resolved conifold in framing f = −1 shown as grey network trajectories. Resolved projections of soliton paths, and the concatenation with their own their logarithmic shifts are also shown (in red and in blue). 28 [PITH_FULL_IMAGE:figures/ful…
Figure 19
Figure 19. Figure 19: Concatenation of a1 and a [+1] 2 for the computation of C12. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_19.png]
Figure 20
Figure 20. Figure 20: Soliton webs aW i for the conormal lagrangian of the trefoil knot in framing f = −1 shown as grey network trajectories. Resolved projections of soliton paths, and the concatenation with their own their logarithmic shifts are also shown. The black dot denotes the punct…
Figure 21
Figure 21. Figure 21: Computation of off-diagonal entries of Cij for the trefoil augmentation curve in framing f = −1. shown as grey network trajectories. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_21.png]
Figure 22
Figure 22. Figure 22: Soliton webs aW i for the conormal lagrangian of the figure eight knot in framing f = 0 shown as grey network trajectories. Resolved projections of soliton paths, and the concatenation with their own their logarithmic shifts are also shown. The black dot denotes the p…
Figure 23
Figure 23. Figure 23: Vi and Vj are shown as red and blue dots respectively. The electric field lines for U(1)i (U(1)j ) are shown in solid red (blue) and the magnetic field lines in dashed red (blue). The green arrow indicates the Poynting vector induced by the interaction between the two…
Figure 24
Figure 24. Figure 24: Two possible monodromies and the corresponding concatenated soliton paths around a loga [PITH_FULL_IMAGE:figures/full_fig_p037_24.png]
Figure 25
Figure 25. Figure 25: Different cases for intersection number +1 around a branch point The generalisation to calculating intersections around higher degree branch points is straighforward, although computationally more challenging since we would now have to take care of the monodromy direc…
Figure 26
Figure 26. Figure 26: Different concatenations for solitons with the same phase. [PITH_FULL_IMAGE:figures/full_fig_p038_26.png]
Figure 27
Figure 27. Figure 27: Soliton-antisoliton webs. D Off-diagonal intersections of basic solitons for the figure-eight aug￾mentation curve We collect details on the computation of Cij for i < j in Figures 28, 29, 30, 31, 32, 33, 34, 35, 36 and 37. 39 [PITH_FULL_IMAGE:figures/full_fig_p039_27.png]
Figure 28
Figure 28. Figure 28: Concatenation of a1 and a [+1] 2 for the computation of C12. -1 [PITH_FULL_IMAGE:figures/full_fig_p040_28.png]
Figure 29
Figure 29. Figure 29: Concatenation of a1 and a [+1] 3 for the computation of C13. 40 [PITH_FULL_IMAGE:figures/full_fig_p040_29.png]
Figure 30
Figure 30. Figure 30: Concatenation of a1 and a [+1] 4 for the computation of C14. -1 [PITH_FULL_IMAGE:figures/full_fig_p041_30.png]
Figure 31
Figure 31. Figure 31: Concatenation of a5 and a [+1] 1 for the computation of C15. 41 [PITH_FULL_IMAGE:figures/full_fig_p041_31.png]
Figure 32
Figure 32. Figure 32: Concatenation of a3 and a [+1] 2 for the computation of C23. +1 [PITH_FULL_IMAGE:figures/full_fig_p042_32.png]
Figure 33
Figure 33. Figure 33: Concatenation of a4 and a [+1] 2 for the computation of C24. 42 [PITH_FULL_IMAGE:figures/full_fig_p042_33.png]
Figure 34
Figure 34. Figure 34: Concatenation of a5 and a [+1] 2 for the computation of C25 [PITH_FULL_IMAGE:figures/full_fig_p043_34.png]
Figure 35
Figure 35. Figure 35: Concatenation of a3 and a [+1] 4 for the computation of C34. 43 [PITH_FULL_IMAGE:figures/full_fig_p043_35.png]
Figure 36
Figure 36. Figure 36: Concatenation of a5 and a [+1] 3 for the computation of C35. -1 [PITH_FULL_IMAGE:figures/full_fig_p044_36.png]
Figure 37
Figure 37. Figure 37: Concatenation of a5 and a [+1] 4 for the computation of C45. 44 [PITH_FULL_IMAGE:figures/full_fig_p044_37.png]

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