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Edge vectors on plabic networks in the disk and amalgamation of totally non-negative Grassmannians

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Pith's one-line read This paper proves that, on boundary-to-boundary plabic graphs in the disk, a Lam system of relations is full rank with totally non-negative image for all positive weights exactly when its edge signature is geometric, and then its boundary…

desk verdict A solid, self-aware completeness theorem for geometric signatures on PBDTP plabic graphs; the main proof holds, but the advertised 'complete' answer is only complete on that subclass and one abstract claim lacks support. read the letter →

arxiv 1908.07437 v4 pith:IEZ7QYFJ submitted 2019-08-20 math.CO

Simonetta Abenda , Petr G. Grinevich This is my paper · ORCID
classification math.CO MSC 14M1505C1005C22
keywords totallynon-negativeGrassmannianspositroidcellsplabicnetworksedgesignaturesgeometricsignatureboundarymeasurementmapamalgamationvectors
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

Lam asked which edge signatures make a system of relations on a plabic graph full rank with a totally non-negative image for every choice of positive weights. The paper answers this for planar bicolored trivalent directed perfect graphs in the disk that are PBDTP, meaning every edge lies on a directed path from the boundary to the boundary. On such a graph the admissible signatures are exactly the geometric signatures, one per graph up to a simple gauge transformation, and the solution at boundary sources is Postnikov's boundary measurement matrix, with image the corresponding positroid cell. This matters because those relation systems are the algebraic transcription of amalgamation, the gluing of small positive Grassmannians that underlies cluster theory and scattering-amplitude computations.

What carries the argument

The central object is the geometric signature: a bit $\epsilon_{U,V}$ assigned to each edge of the oriented network, computed from two geometric indices — the local winding number of an ordered pair of edges around a fixed gauge ray direction $l$, and the number of intersections of the edge with gauge rays emitted from the boundary sources. The Lam system writes these bits into relations $z_{U,e}=(-1)^{\epsilon_{U,V}}w_{U,V}z_{V,e}$ at edges, equality at black vertices, and a zero sum at white vertices. The signature carries the whole argument because Theorem 7.16 says it is the only choice, up to gauge equivalence, that guarantees full rank and total non-negativity for all positive weights; its face-level counterpart (Theorem 7.23) says the total signature around any face is just the parity of the number of white vertices on it.

What would settle it

Test the sign obstruction in equation (7.16) on a small PBDTP graph: pick a signature that differs from the geometric one in parity on some boundary-to-boundary path or some conservative cycle, and choose positive weights concentrated on that path or cycle with all other weights of order $\delta \ll 1$. The theorem predicts that some boundary matrix entry, or its denominator, changes sign and breaks total non-negativity for sufficiently small $\delta$; finding a non-geometric signature whose matrix stays totally non-negative for all positive weights on such a graph would disprove Theorem 7.16.

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Extended reading notes

Core claim

The central claim, Theorem 7.16, is a complete characterization. Let $G$ be a PBDTP plabic graph representing an irreducible positroid cell $S^{TNN}_M$. A signature $\epsilon_{U,V}$ on the edges induces a Lam system of relations that has full rank and a totally non-negative image for every choice of positive weights if and only if $\epsilon_{U,V}$ is equivalent to the geometric signature; in that case the boundary-source solution is exactly the Postnikov boundary measurement matrix and the image is $S^{TNN}_M$. The geometric signature is unique up to gauge equivalence, and the paper shows that changes of orientation, gauge-ray direction, and vertex position act as gauge transformations of the signature. The paper also gives rational formulas for the edge-vector components at internal edges, extending Talaska's flow formula to the interior of the graph.

Load-bearing premise

The load-bearing premise is the PBDTP condition: every edge of the plabic graph must lie on at least one directed path from the boundary back to the boundary; if any edge fails this, path and cycle parities no longer determine a signature and the 'if and only if' statement gains extra gauge freedom.

Editorial extensions

If this is right

  • Lam's program for plabic networks in the disk is settled: the signatures that preserve total non-negativity under amalgamation of $\mathrm{Gr}^{TP}(1,3)$ and $\mathrm{Gr}^{TP}(2,3)$ are precisely the geometric ones.
  • The boundary measurement matrix is recovered as the unique solution of the geometric linear system at boundary sources, so internal edge vectors give a consistent extension of Postnikov's and Talaska's boundary formulas into the graph.
  • On acyclically orientable graphs, all edge-vector components are subtraction-free rational functions of the positive weights, and null edge vectors cannot occur.
  • For graphs that are not PBDTP, the completeness statement must be modified; the extra gauge freedom at edges not lying on boundary-to-boundary paths is untrivial and can even support null edge vectors on reducible networks.
  • The face-level signature formula connects the geometric signature to Kasteleyn and dimer sign conditions, and the paper identifies its master signature on Le-networks as geometric.

Reading between the lines

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

  • If Theorem 7.16 is right, it gives a practical criterion: to test whether a proposed relation system on a disk plabic network is admissible, one only needs to compare path and cycle parities with the geometric signature, avoiding direct checks over all positive weights.
  • The PBDTP condition may be the right generality for a slightly weaker statement: without it, signatures are still constrained on boundary-to-boundary paths and cycles, but the leftover edge gauge degrees of freedom likely form a finite-dimensional family; classifying that family would extend the theorem to arbitrary plabic graphs.
  • The face-signature formula suggests a direct bridge to dimer models: one could try to realize the geometric signature explicitly as a Kasteleyn sign matrix on any PBDTP graph, not only on reduced bipartite graphs, and use it to count perfect matchings with boundary conditions.
  • A natural testable extension, mentioned as an open problem in the paper, is the same construction on networks in the annulus or on other surfaces with boundary; the gauge-ray machinery should adapt with a modified Talaska formula, but the signature classification may pick up new topological invariants.
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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 develops a theory of edge vectors on planar bicolored directed trivalent perfect (plabic) networks in the disk that parametrize totally non-negative positroid cells. The central construction assigns to each edge a vector whose components are signed sums over directed paths from that edge to boundary sinks, with signs determined by a winding index relative to a gauge ray direction and by intersections with gauge rays. The authors prove that the resulting vertex relations have full rank (Theorem 3.11), that the edge-vector components are rational functions of the weights with subtraction-free denominators and explicit flow expansions (Theorem 3.12), and that the edge vectors transform predictably under changes of orientation, gauge ray direction, vertex and weight gauge, and Postnikov moves. In Section 7 the vertex relations are recast as Lam's half-edge relations, and geometric signatures are introduced. The main completeness result, Theorem 7.16, states that for a PBDTP graph representing an irreducible positroid cell, a signature gives a full-rank system with totally non-negative image for every choice of positive weights if and only if it is equivalent to the geometric signature, in which case the boundary solution is the Postnikov boundary measurement matrix and the image is the expected positroid cell. The paper also gives a face formula for geometric signatures and an interpretation of the relations as totally non-negative amalgamation of small Grassmannians.

Significance. If Theorem 7.16 is correct, it answers Lam's signature question for the PBDTP subclass of plabic networks and gives a concrete bridge among geometric relations, boundary measurement matrices, and amalgamation of totally non-negative Grassmannians. The paper's strengths are its explicit and constructive character: the full-rank statement in Theorem 3.11 is proved by identifying the determinant with the sum of conservative-flow weights, the edge-vector components are given by closed Talaska-type flow formulas, and the invariance and vertex-consistency checks are carried out in the appendices. The main caveat is that the advertised completeness is proved only under the PBDTP hypothesis, and the relation between this hypothesis and the stated claim of completeness needs to be made precise in the published version.

major comments (3)
  1. [Definition 2.5; Lemma 7.14; Remark 7.15; Theorem 7.16] The completeness theorem is proved only for PBDTP graphs, and the PBDTP condition is load-bearing in Lemma 7.14: the step in which equal parities on all boundary-to-boundary paths and cycles are converted into gauge equivalence of signatures uses the fact that every edge lies on such a path. Remark 7.15 explicitly concedes that outside this class there is extra gauge freedom and that the statement of Theorem 7.16 must be modified. The abstract and the introduction nevertheless advertise a 'complete and explicit characterization' and describe the setting as 'optimal.' Since the paper gives no concrete non-PBDTP graph for which a non-geometric signature still has full rank and totally non-negative image for all positive weights, it does not establish that PBDTP is the optimal domain. The authors should either provide such an example or an argument that none exists, or rephrase the advertised completeness claim as a characterization on the PBDTP subclass.
  2. [Abstract; Section 7.3 (Remark 7.24, Conjecture 7.25)] The abstract states that 'the image of the boundary measurement map and the dimer partition functions do not coincide if the graph is not bipartite.' I could not find this assertion stated or proved anywhere in the body. Section 7.3 only conjectures a Kasteleyn-type interpretation for PBDTP graphs and notes that for reduced bipartite graphs the geometric signature realizes Speyer's variant; no non-bipartite comparison is proved. This claim should be removed from the abstract or supported by a proof in the text.
  3. [Theorem 7.16, Step 3 (Eqs. (7.18)–(7.19))] The asymptotic sign argument in Step 3 of the proof of Theorem 7.16 draws conclusions about the sign of the matrix entry A_{ij} from total non-negativity. Total non-negativity is a condition on maximal minors, not directly on individual entries of the reduced row echelon form. The argument should explicitly use the fixed-sign relation between A_{ij} and the maximal minor obtained by replacing the pivot column i_r with column j, or else state the sign convention that makes this implication immediate. This is a local proof gap rather than an error in the statement, but it is part of the proof of the central theorem and should be repaired.
minor comments (4)
  1. [Abstract and Section 1] The phrase 'complete and explicit characterization' appears before the PBDTP hypothesis is introduced; state the hypothesis at the first mention of completeness so that the scope of the theorem is visible to the reader.
  2. [Section 1] The word 'optimal' in the description of the PBDTP setting is stronger than what is proved. Replacing it with 'the setting in which we prove completeness' would better match the content of Theorem 7.16 and Remark 7.15.
  3. [Example 3.15 and Section 6] Example 3.15 gives a clear demonstration that null edge vectors can occur on reducible networks. It would help the reader if the example were referenced again at the beginning of Section 6, where the phenomenon is discussed systematically.
  4. [Definition 3.4 and Eq. (3.5)] The notation int(e) in Eq. (3.5) refers to the edge incident to a boundary sink, but the same symbol is used earlier for the intersection number of an arbitrary edge in a path. Clarify that the boundary edge has its own intersection number, since the distinction is used in several later formulas.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the completeness theorem is proven from Postnikov and Talaska benchmarks, with the PBDTP restriction explicitly flagged as a limitation rather than smuggled in.

full rationale

The paper's central claim, Theorem 7.16, is an if-and-only-if statement: a signature on a PBDTP graph yields full-rank Lam relations with totally non-negative image for all positive weights exactly when it is equivalent to the geometric signature. The forward direction follows from Theorem 7.7, which re-expresses the previously constructed edge-vector system in Lam form and identifies the boundary solution with the Postnikov boundary measurement matrix; this uses external results of Postnikov and Talaska as benchmarks. The reverse direction is proved directly in Steps 1-4 of Theorem 7.16: after the sign-weight substitution (7.14), the Talaska-type formula (7.16) is used to show that total non-negativity forces the parity of the signature on every boundary-to-boundary path and every one-component conservative flow to match the geometric signature, and then Lemma 7.14 converts this parity coincidence into gauge equivalence. The key reduction (7.18) and (7.19) is a path-weight dominance argument, not an assumption of the conclusion. The PBDTP hypothesis is expressly stated as essential in Remark 7.15: edges not on any boundary-to-boundary path never enter the boundary measurement and admit extra gauge freedom, so the theorem is honestly scoped rather than circular. The paper does contain self-citations, notably to the authors' prior work [4] for the master signature on Le-networks, but that citation is not load-bearing for the completeness proof: the same section gives a direct construction by drawing gauge rays almost horizontally so that the master signature coincides with the geometric signature. One non-circular defect is that the abstract claims that boundary-measurement images and dimer partition functions do not coincide for non-bipartite graphs, while the body does not appear to provide a proof or location for this claim; this is a missing-support/correctness issue, not a circularity issue.

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

No parameters are fitted to data: the central construction depends on choices (orientation O, gauge ray direction l, positive weights) that are shown to be gauge-equivalent for the signature. The only non-standard objects introduced are the geometric signature and edge/half-edge vectors, which are mathematical constructs with explicit definitions, not postulated entities requiring independent falsifiable evidence.

assumptions (5)
  • domain assumption Graphs are planar bicolored directed trivalent perfect graphs in the disk with boundary vertices on a common interval and each boundary edge attached to an internal bivalent white vertex (Definition 2.3, Remark 2.4).
    Restricts the class of networks; the common-interval placement is used to define gauge rays and winding numbers, and trivalency is claimed to be non-restrictive.
  • domain assumption PBDTP condition: every edge belongs to at least one directed path from boundary to boundary (Definition 2.5).
    Essential for the parity equivalence Lemma 7.14 and the completeness Theorem 7.16; without it extra signature gauge freedom appears.
  • domain assumption The gauge ray direction l is generic: no internal edge is parallel to l and no boundary gauge ray passes through an internal vertex (Definition 3.1).
    Generic position is needed to define the local winding and intersection indices; the authors argue results are independent of l up to gauge.
  • domain assumption Edge weights are positive real numbers, and for acyclically orientable networks they are expressed in a canonical basis.
    Total non-negativity and the subtraction-free rational formulas require positive weights; this is the standard setting of Postnikov's boundary measurement map.
  • standard math Background results from Postnikov's boundary measurement map, Talaska's flow formula for Plücker coordinates, and Lam's amalgamation framework are taken as established.
    The paper adapts these results to internal edges and uses them as benchmarks, e.g., Theorem 3.12 is an adaptation of Talaska [57].

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Pith. "Pith review of Edge vectors on plabic networks in the disk and amalgamation of totally non-negative Grassmannians." pith.science (2026). https://pith.science/paper/IEZ7QYFJ

@misc{pith2026190807437,
  author       = {Pith},
  title        = {Pith review of: Edge vectors on plabic networks in the disk and amalgamation of totally non-negative Grassmannians},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IEZ7QYFJ}},
  note         = {Machine review of arXiv:1908.07437}
}
abstract

Amalgamation in the totally non-negative part of positroid varieties is equivalent to gluing copies of $Gr^{TP}(1,3)$ and $Gr^{TP}(2,3)$. Lam has proposed to represent amalgamation in positroid varieties by equivalence classes of relations on bipartite graphs and identify total non-negativity via edge signatures. Here we provide an explicit characterization of such signatures on the planar bicolored trivalent directed perfect networks in the disk parametrizing positroid cells $S_M^{TNN}$.To a graph $G$ representing $S_M^{TNN}$, we associate a geometric signature satisfying full rank condition and total non--negativity. Such signature is uniquely identified by geometric indices ruled by orientation and gauge ray direction. The image of this map coincides with that of Postnikov boundary measurement map. We solve the system of geometric relations generalizing Postnikov's and Talaska's results for the boundary edges to the internal edges of the graphs: the edge vector components are rational in the weights with subtraction--free denominators, and have explicit expressions in terms of conservative and edge flows. At boundary sources the edge vectors give the boundary measurement matrix. If $G$ is acyclically orientable, all components are subtraction-free rational in the weights w.r.t. a convenient basis. We provide explicit formulas for the transformation rules w.r.t. changes the orientation, the several gauges of the given network, moves and reductions of networks. We show that the image of the boundary measurement map and the dimer partition functions do not coincide if the graph is not bipartite.

Figures

Figures reproduced from arXiv: 1908.07437 by the authors.

Figure 1
Figure 1. The rays starting at the boundary sources for a given orientation of the network uniquely fix the edge vectors. Definition 3.1. The gauge ray direction l. A gauge ray direction is an oriented direction l with the following properties: (1) The ray l starting at a boundary vertex points inside the disk; (2) No internal edge is parallel to this direction; (3) All rays starting at boundary vertices do not contain intern… view at source ↗
Figure 2
Figure 2. The local rule to compute the winding number. π-ϵ a) b) -π+ϵ V V c) d) π-ϵ -π+ϵ V V [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. If the ordered pair (ek, ek+1) is antiparallel at V , we slightly rotate the two edge vectors at V to compute wind(ek, ek+1). Using (3.3) and (3.2), we get a): wind(ek, ek+1) = 1; b): wind(ek, ek+1) = −1; c): wind(ek, ek+1) = 0; d): wind(ek, ek+1) = 0. Let bir , r ∈ [k], bjl , l ∈ [n − k], respectively be the set of boundary sources and boundary sinks associated to the given orientation. Then draw the rays lir , r ∈… view at source ↗
Figures from the paper (33 more)
Figure 4
Figure 4. Figure 4: The graph of Remark 3.6. first edge loop it makes; more precisely, given all pairs l, s with s > l and el = es, one chooses the one with the smallest value of s and removes the cycle Vl el → Vl+1 el+1 → Vl+2 → . . . es−1 → Vs, from P. Remark 3.6. An edge loop-erased wa…
Figure 5
Figure 5. Figure 5: The linear system at black and white vertices as a function of the sector Si in which the gauge ray direction l is located. (1) At each bivalent vertex with incoming edge e and outgoing edge f: (3.9) Ee = (−1) int(e)+wind(e,f)weEf ; (2) At each trivalent black vertex w…
Figure 6
Figure 6. Figure 6: We illustrate the proof of Corollary 3.13. The path P is the union of black and green edges, the path P˜ is the union of black and blue edges. Additional path Q is drawn magenta, and the edge ej,ir is drawn gold. On the left (respectively right) the gauge ray starting …
Figure 7
Figure 7. Figure 7: The computation of edge vectors using Theorem 3.12 and Lemma 4.4. The path along which we change orientation is colored red in both Figures; we mark regions to compute the indices in (4.4) and (4.5) [left]. Next, let us add a directed path Q from bj to bir very close t…
Figure 8
Figure 8. Figure 8: We illustrate Proposition 4.1. at up. Similarly Eu1 = (1, 0) since there is only one loop erased walk and three edge flows from u1. Finally the representative matrix associated to this system of vectors is A[1] = ( 1+2p 1+p+q , 1). 4. Dependence of edge vectors on orie…
Figure 9
Figure 9. Figure 9: We illustrate the marking of the regions. divided into two arcs, each oriented from j0 to i0. Then we mark a region with a + if its boundary is oriented, otherwise mark it with − (see [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: The rule of the sign at ei0 [PITH_FULL_IMAGE:figures/full_fig_p023_10.png]
Figure 11
Figure 11. Figure 11: The effect of the weight gauge transformation at a white [left] and at a black [right] vertex on the edge vectors. Remark 4.8. The weight gauge freedom [48]. Given a point [A] ∈ S TNN M and a planar directed graph G in the disk representing S TNN M , then [A] is repre…
Figure 12
Figure 12. Figure 12: The vertex gauge transformation at a white [left] and at a black [right] vertex consists in moving an internal vertex from position V to V˜ . (1) Let Ee and E˜ e respectively be the system of edge vectors on (N ,O, l) and on (N˜ ,O, l), where N˜ is obtained from N mov…
Figure 13
Figure 13. Figure 13: The effect of the square move. 5. Effect of moves and reductions on edge vectors In [48] it is introduced a set of local transformations - moves and reductions - on planar bicolored networks in the disk which leave invariant the boundary measurement map. Two networks …
Figure 14
Figure 14. Figure 14: The insertion/removal of an unicolored internal vertex is equivalent to a flip move of the unicolored vertices. F˜ 2 = α1(−1) int(h1)+γ2(h4)+wind(h2,−h4)+wind(h3,h4)+1F3+α2α −1 4 (−1) 1+int(h2)+int(h4)+γ2(h4)+wind(h2,−h4)F4. F˜ 1 = α˜1(−1) int(h1)+wind(−h1,h2)F˜ 2, F˜…
Figure 15
Figure 15. Figure 15: The middle edge insertion/removal. ~ ~ ~ ~ [PITH_FULL_IMAGE:figures/full_fig_p028_15.png]
Figure 16
Figure 16. Figure 16: The parallel edge reduction. Therefore, additivity of the winding numbers holds in this special case wind(ei , e0)+wind(e0, ej) = wind(ei , ej), with ei – any incoming vector, ej – any outgoing vector involved in the move. Finally, F˜ i = Fi in all cases, E˜ 0 = { E0,…
Figure 17
Figure 17. Figure 17: Left: the dipole reduction. Right: the leaf reduction. q p u v w V4 s (1+p+s+2sp)/q u v w p/(1+q+p) ~ ~ ~ [PITH_FULL_IMAGE:figures/full_fig_p029_17.png]
Figure 18
Figure 18. Figure 18: The appearance of null vectors on reducible networks [left] and their elim￾ination using the gauge freedom for unreduced graphs of Remark 4.9 [right]. (R3) The leaf reduction The leaf reduction occurs when a network contains a vertex u incident to a single edge e1 end…
Figure 19
Figure 19. Figure 19: The reformulation of the linear relations for the edge vectors as geometric relations for half-edge vectors compatible with Lam [41] approach at the vertices [top] and at edges [bottom]. We use the abridged notations wind(i, j) ≡ wind(ei , ej ) and int(i) ≡ int(ei). g…
Figure 20
Figure 20. Figure 20: The geometric signature for Example 7.9. sources. The reason of this choice is that the total geometric signature at each face is invariant with respect to changes of orientations of the graph. We remark that the above system is meaningful also in a perfectly oriented…
Figure 21
Figure 21. Figure 21: On the left: a simple path P1 from the boundary source bi to the boundary sink bj passing through the edge e. On the right: a one-component conservative flow C = P2 + P3 containing e and a simple path P = P1 + P2 + P4 from the boundary source bi to the boundary sink b…
Figure 22
Figure 22. Figure 22: The topological winding wΩ at pairs of edges at internal vertices with change of orientation (top left), at the boundary sinks (top right) and at the boundary sources (bottom). Next we define the total winding of a face Ω: (7.21) wind(Ω) = l ∑ i=1 windΩ(vi). If Ω is a…
Figure 23
Figure 23. Figure 23: The Young diagram associated to the partition (5, 5, 4, 2), k = 4, n = 9. 𝒘𝟏𝟗 𝒘𝟏𝟔 𝒘𝟏𝟓 𝒘𝟐𝟑 𝒘𝟒𝟗 𝒘𝟒𝟔 𝒘𝟒𝟓 𝒘𝟕𝟖 𝒘𝟒𝟖 𝟎 𝟎 𝟎 𝟎 𝒘𝟐𝟓 𝟎 𝟎 𝟏 𝟒 𝟕 𝒘𝟐𝟓 𝒘𝟐𝟑 𝒘𝟕𝟖 𝒘𝟒𝟗 𝒘𝟒𝟖 𝒘𝟒𝟔 𝒘𝟒𝟓 𝒘𝟏𝟗 𝒘𝟏𝟔 𝒘𝟏𝟓 𝒃𝟗 𝒃𝟖 𝒃𝟕 𝒃𝟔 𝒃𝟓 𝒃𝟒 𝒃𝟑 𝒃𝟐 𝒃𝟏 𝑽𝟏𝟓 𝑽𝟏 𝑽𝟐𝟓 𝑽 𝑽𝟏𝟔 𝟏𝟗 𝑽 𝑽𝟒𝟖 𝑽𝟒𝟔 𝟒𝟗 𝑽𝟕𝟖 𝑽𝟕 𝑽𝟒𝟓 𝑽𝟒 𝑽𝟐𝟑 𝑽𝟐 0 0 0 0 1 1 1 1…
Figure 24
Figure 24. Figure 24: Left: a Le–tableau T for the Young diagram of [PITH_FULL_IMAGE:figures/full_fig_p044_24.png]
Figure 25
Figure 25. Figure 25: Amalgamation of GrTP(2, 4) using the definition [left] and the linear system for half edge vectors of Theorem 7.7 [right]. of on–shell scattering processes in terms of simpler on-shell diagrams [8, 9]. In the case of Grassmannians it may be explicitly described as the…
Figure 26
Figure 26. Figure 26: Configurations at black [left] and white [right] vertices when e1, e2 belong to P0 or to Q0. (1) If the orientation changes along a cycle Q0, then int(f) = int ̂(f) and the starting and the ending point of f belong to the same region as V , therefore γ(f) = γ(g); (2) …
Figure 27
Figure 27. Figure 27: On the left e1 and e2 belong to the same half-plane, γ2(e1) = γ2(e2) = 1, γ2(−e1) = γ2(−e2) = 0, wind(e1, e2) = wind(−e2,−e1) = 0. On the right e1 and e2 belong to opposite half-planes, γ2(e1) = γ2(−e2) = 0, γ2(−e1) = γ2(e2) = 1, wind(e1, e2) = 0, wind(−e2,−e1) = 1. e…
Figure 28
Figure 28. Figure 28: Cyclic order on triples of vectors. By definition [e, f, g] = [f, g, e] = [g, e, f] = 1 − [e, g, f] = 1 − [g, f, e] = 1 − [f, e, g]. Indeed, if e1 and e2 belong to the same half-plane with respect to l, then γ2(e1) + γ2(e2) = 0 ( mod 2) and wind(e1, e2) = wind(−e2,−e1…
Figure 29
Figure 29. Figure 29: Check of (A.15), (A.17) for one configurartion. (2) If V is white, then: (A.16) γ(f) + γ1(e1) = [e1,−e2,−f] (mod 2), (A.17) wind(e1, f) + wind(−e2, f) + wind(−e2,−e1) + γ2(e1) = 1 − [e1,−e2,−f] (mod 2). Proof. In the proof all identities hold mod 2. To prove (A.14), l…
Figure 30
Figure 30. Figure 30: Appendix B. Proof of the invariance of geometric signature Remark B.1. All the identities in this Section are to be treated mod 2. (1) Let us prove that every change of orientation corresponds to a gauge transformation defined by (7.9) (a) If U and V do not belong to …
Figure 31
Figure 31. Figure 31: Using (B.2) we get (f) + ˆ(f) = int(f) + int ̂(f) + wind(e1, e2) + 1 + γ2(e2) + [e1,−e2,−f] = = [int(f) + int ̂(f) + γ1(e1) + [e1,−e2,−f]] + [1 + γ1(e1) + γ2(e1) + wind(e1, e2)] = η(U) + η(V ). (c) If both U and V belong to Q (P respectively) and none of them is a b…
Figure 32
Figure 32. Figure 32: (iv) If both U and V are white (Case 4), then (e2) + ˆ(e2) = int(e2) + int ̂(e2) + wind(e1, e2) + wind(−e2,−e3). Using (B.2) and (B.3) we obtain (e2) + ˆ(e2) = γ1(e1) + γ1(e2) + wind(e1, e2) + wind(e2, e3) + γ2(e2) + γ2(e3) = = [wind(e1, e2) + γ1(e1) + γ2(e2) + 1]…
Figure 33
Figure 33. Figure 33: (c) Let us check what happens if the gauge ray direction is parallel (not antiparallel) to one of the edges at U or V . In [PITH_FULL_IMAGE:figures/full_fig_p054_33.png]
Figure 34
Figure 34. Figure 34: If one of the edges at U becomes parallel to l, some windings may change. We mark the edges participating in these windings with continuous lines and all other edges with dashed lines. (vii) If U, V are black, and l passes the direction of e3, then we have the followi…
Figure 35
Figure 35. Figure 35: If one of the edges at U becomes parallel to l, some windings may change. We mark the edges participating in these windings with continuous lines and all other edges with dashed lines. calculation shows that the statement is correct since: (f) − ˜(f) = 0, (gi) − ˜…
Figure 36
Figure 36. Figure 36: If one of the edges at U becomes parallel to l, some windings may change. We mark the edges participating in these windings with continuous lines and all other edges with dashed lines. If W = b is a boundary source, and U does not pass the gauge ray starting at b, the…

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