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Monodromies of surfaces in 3-manifolds, right-veeringness, and primeness of links

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Pith's one-line read This paper proves that alternative links are visually prime: if a non-prime link has an alternative diagram, the decomposition of the link as a connected sum is visible in that diagram.

desk verdict Strong new framework and likely-true visual primeness theorem, but the proof of Theorem 6.2 has a real gap in the final reduction that needs fixing. read the letter →

arxiv 2509.09615 v1 pith:DN6RDIY3 submitted 2025-09-11 math.GT

Peter Feller , Lukas Lewark , Miguel Orbegozo Rodriguez This is my paper · ORCID
classification math.GT MSC 57K1057R17
keywords partialmonodromyincompressiblesurfacesMurasugisumsright-veeringstronglyquasipositivevisualprimenessalternativelinksCromwell'sconjecture
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 extends the classical monodromy of fiber surfaces and open books to every incompressible surface in a 3-manifold, where the monodromy becomes a partially defined self-map on isotopy classes of arcs. It proves a composition formula for Murasugi sums—the monodromy of the sum is the composition of the summand monodromies—and uses veering properties of arcs to detect when fixed arcs in a sum must come from fixed arcs in the summands. On this basis it shows that strongly quasipositive surfaces are right-veering, giving a contact-geometry-free proof. The payoff is a proof that alternative links are visually prime: if a non-prime link has an alternative diagram, the diagram contains an honest decomposition circle. This unifies and extends prior visual-primeness results for alternating links, positive links, and positive braids, and is a step toward Cromwell's conjecture.

What carries the argument

The load-bearing object is the partial monodromy φΣ: A(Σ) → A(Σ), a partially defined self-map on the set of isotopy classes of properly embedded arcs of an incompressible surface, defined by product disks. The key identity is the Murasugi-sum composition formula φ = Φ1 ∘ Φ2, where Φi is the extension by the identity of the summand monodromy to the summed surface. Right-veeringness (every arc that goes to the other side satisfies a ≤ φ(a) in the boundary order) and left-veeringness are the veering properties that make the fixed-arc criterion work: in a tree-guided Murasugi sum with oppositely veering summands, fixed arcs restrict to fixed arcs (Theorem 4.2), and a surface built from strictly

What would settle it

Find a non-prime link that admits an alternative diagram with no honest decomposition circle; equivalently, run a computer search over alternative diagrams of connected sums up to moderate crossing number and check whether every such diagram has a visible decomposition circle. A single counterexample would falsify Theorem 6.2, and a Murasugi sum of two strictly right-veering incompressible surfaces along an essential region that has an essential fixed arc would falsify Proposition 3.13 and Theorem 4.2.

Watch

Extended reading notes

Core claim

The paper's central claim is that monodromy data is available not only for fibered links but for all incompressible surfaces. The authors define the partial monodromy φΣ of an incompressible surface Σ in a 3-manifold as the partial self-map of the arc set A(Σ) that sends an arc a to the arc φ(a) on the other side of a product disk, if such a disk exists. For a Murasugi sum Σ of Σ1 and Σ2, they prove φ = Φ1 ∘ Φ2, where Φi extends φi by the identity, recovering the classical fibered composition formula as a special case. From this composition law they derive a primeness criterion (Proposition 3.10 and Theorem 4.2): in a tree-guided Murasugi sum of right-veering and left-veering summands, any f

Load-bearing premise

The proof of Theorem 6.2 relies on an unproven geometric assertion: a fixed essential arc in a vertex surface of the tree can be assumed to decompose that surface into a disk plus some of the boundary-connected-sum arcs; if the arc instead winds through the summands in a more complicated way, the constructed honest decomposition circle might not exist.

Editorial extensions

If this is right

  • Alternative links are visually prime: any alternative diagram of a non-prime link contains an honest decomposition circle (Theorem 6.2).
  • This recovers visual primeness of alternating links and positive links, and positive braids, as special cases; it also recovers the homogeneous-braid visual primeness result, since homogeneous braid diagrams are alternative.
  • Strongly quasipositive surfaces in S^3 are right-veering; they are strictly right-veering exactly when they are irreducible (Proposition 5.1).
  • A family of incompressible surfaces closed under positive Hopf plumbing is right-veering unless it contains a surface with an essential 0-framed unknotted annulus (Theorem 5.3).
  • All arborescent links are prime, and the tight/right-veering characterization of contact structures is reproved with minimal contact geometry (Corollary 5.5).

Reading between the lines

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

  • If the tree decomposition of alternative diagrams is the template, one can test Cromwell's conjecture more broadly by seeking tree decompositions of arbitrary diagrams into veering pieces; the paper's approach suggests that visual primeness is a veering phenomenon, not a fiberedness phenomenon.
  • The partial-monodromy formalism may transfer other open-book tools—for instance, detecting homotopy ribbonness or considering arc complexes—to non-fibered surfaces, since the arc set is the only bookkeeping needed.
  • The right-veering characterization (Theorem 5.3) offers a concrete search target: in any family of surfaces closed under positive Hopf plumbing, the first obstruction to right-veeringness is an essential 0-framed unknotted annulus; one could look for such annuli in families arising from other diagram classes.
  • The proof of Proposition 5.1 converts a left-veering arc into a sobering arc with no interior intersections by positive Hopf plumbings; this suggests a constructive algorithm to exhibit overtwisted disks from non-right-veering open books, which could be made explicit.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper develops a theory of partial monodromies for arbitrary incompressible surfaces in 3-manifolds, generalizing the monodromy of fiber surfaces in open books. The main technical results are a composition formula for partial monodromies under Murasugi sums (Lemma 3.1), a primeness criterion for fixed arcs in tree-guided Murasugi sums (Theorem 4.2), a proof that strongly quasipositive surfaces are right-veering without contact geometry (Proposition 5.1), and a characterization of right-veering families closed under positive stabilization (Theorem 5.3). These tools are applied to prove that alternative link diagrams are visually prime (Theorem 6.2), subsuming previous visual-primeness results for alternating, positive, and homogeneous braid diagrams. The paper is ambitious and contains several substantial contributions, but the proof of the headline application has significant gaps.

Significance. If the results are correct, this is a meaningful step forward: the partial-monodromy framework extends open-book technology to all incompressible surfaces, and the visual-primeness theorem for alternative links would be a notable advance on Cromwell's conjecture. The paper also gives a new, topological proof that strongly quasipositive surfaces are right-veering and offers an alternative route to the Honda–Kazez–Matić tightness characterization. The composition formula and the primeness criteria are likely to be useful tools. However, the main application rests on a proof that is incomplete at a load-bearing point, and there is a false statement in the definitional discussion of alternative diagrams, so the significance is conditional on a successful revision.

major comments (4)
  1. [§6, proof of Theorem 6.2] The assertion “Then, by Proposition 3.7, a′ cuts out a disk D together with a subcollection of the sj” does not follow. Proposition 3.7 only states that a fixed arc in a boundary connected sum restricts to fixed arcs in each summand; it does not describe how those restrictions assemble globally. An essential fixed arc could in principle pass through several summands and connecting bands, with each restriction boundary-parallel, without cutting off a single disk together with some s_j. The required outermost-arc/innermost-disk argument is absent. Since the honest decomposition circle is eventually constructed from this structure, this gap is load-bearing.
  2. [§6, proof of Theorem 6.2] The step “b′ is obtained from b by a free isotopy along the boundary, so b′ is also fixed in Σv′” is unjustified. Fixedness (Definition 2.6) is defined for isotopy classes of arcs with endpoints fixed pointwise, and a free isotopy slides endpoints along ∂Σ, changing the arc class. The partial monodromy is not shown to be invariant under such endpoint-sliding. Therefore the conclusion that b′ is fixed in Σv′ does not follow from the fixedness of b. This affects the construction of c = b′, which is one of the two ways the proof aims to produce an honest decomposition circle.
  3. [§6, Definition 6.1 and following paragraph] The statement “Dv is alternating if and only if it is positive or negative” is false. The closure of the positive 3-braid (σ1σ2)^4 is special (all Seifert circles are non-separating) and positive, but it is the (3,4) torus knot, which is not alternating. Thus the claimed equivalence between alternating and positive/negative special diagrams is not correct. This matters because the definition of alternative links in the paper is justified by this equivalence, and the equivalence to Kauffman's original definition is used to assert that the class studied is genuinely the class of alternative links. The authors need to correct this statement or adjust the definition of alternative diagrams accordingly.
  4. [§6, proof of Theorem 6.2] The sentence “Since b′ ⊂ D, the discussion in Remark 2.5 implies that b′ is fixed in Σv” is not justified. Remark 2.5 explains how a partial monodromy can be recovered from a cutting system of arcs; it does not state that an arbitrary arc contained in a disk cut out by fixed arcs is itself fixed. A product disk witnessing fixedness of b′ would need to be constructed separately. This step is used to conclude that b′ is fixed in both Σv and Σv′, so it is essential to the proof.
minor comments (4)
  1. [§2.3, first paragraph] The first line reads “For 3-manifolds M1 and M3”; this appears to be a typo for M1 and M2.
  2. [§3.2, proof of Proposition 3.10] The phrase “W.l.o.g. the endpoints of a are not on ∂Σ1 ∩ ∂Σ2 (if not, isotope them in the boundary, do the argument, and isotope back)” is another instance where endpoint-sliding is used without justifying invariance of fixedness. This may be repairable, but it needs an explicit argument.
  3. [§5, statement of Theorem 5.3 and Remark 5.4] The introduction advertises a characterization (“if and only if”) of right-veering families, but Theorem 5.3 as stated gives only a sufficient condition in terms of 0-framed unknotted annuli; the full characterization appears only in Remark 5.4 via property (2). The presentation would be clearer if the theorem statement matched the stronger characterization or if the remark were clearly flagged as the converse.
  4. [§6, proof of Lemma 6.5] In the proof of Lemma 6.5(i), the statement “A split union of links is prime if and only if all of the individual links are” is a standard fact, but the wording could be misinterpreted: a split union with two nontrivial components is not prime, and the equivalence should be stated with the convention for prime links. Please clarify the convention.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: a minor self-cited combinatorial lemma is used, but the main results are derived from stated geometric hypotheses rather than assumed.

full rationale

The paper's derivation chain is not circular. The partial monodromy and the composition formula (Lemma 3.1) are proved from product-disk geometry rather than assumed; the formula is not a definitional identity. Proposition 3.10's proof uses Lemma 3.12, cited to the authors' own [FLO24, Lemma 3.7], but this is a parameter-free chord-combinatorics statement whose assumptions do not include the target results; under the review rules it counts as independent support. Theorem 4.2 is an induction on trees using Proposition 3.10; Proposition 5.1 is obtained by an explicit positive-plumbing construction, not by fitting or renaming. Theorem 6.2's base case is Lemma 6.4, attributed to external work [Men84, Oza02, BG24], and the rest reduces fixed arcs to vertex surfaces via Theorem 4.2. The only concerns are proof-completeness, not circularity: the assertion in the proof of Theorem 6.2 that a fixed essential arc a' in a boundary-connected-sum vertex surface cuts out a disk with a subcollection of the s_j is not fully justified by the cited Proposition 3.7, and the boundary-isotopy step preserving fixedness is terse. These are gaps or missing details; they do not make the conclusion equal to an input by construction. No fitted parameter is renamed a prediction, and no uniqueness theorem from the authors' prior work is imported to force the choice. The score reflects one minor self-citation in a key lemma, but with independent content.

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

No free parameters or invented entities appear. The paper's contribution is a new framework (partial monodromies) rather than new postulated objects; all substantive background is cited from the prior literature.

assumptions (6)
  • domain assumption Product disks exist for incompressible surfaces and define the partial monodromy uniquely (Remark 2.2)
    Foundation of the whole paper; asserted to follow from incompressibility via Gabai's product disk theory, not proven here.
  • domain assumption Murasugi sums preserve incompressibility (Gabai [Gab83b, Gab85])
    Used in Section 4 to iterate Murasugi sums along trees.
  • standard math The chord lemma [FLO24, Lemma 3.7]
    Combinatorial lemma from the authors' prior paper, used in Proposition 3.10.
  • domain assumption Rudolph's characterization of strongly quasipositive surfaces as full subsurfaces of positive torus link fibers
    Used to connect the family of strongly quasipositive surfaces to the abstract condition in Theorem 5.3.
  • domain assumption Special alternating links are visually prime [Men84, Oza02, BG24] (Lemma 6.4)
    Base case for the visual primeness theorem; taken from the literature.
  • domain assumption Goodman's sobering arcs yield overtwisted disks, and Eliashberg's classification of overtwisted contact structures
    Used in Corollary 5.5 to connect the new results to the Honda-Kazez-Matic tightness characterization.

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Pith. "Pith review of Monodromies of surfaces in 3-manifolds, right-veeringness, and primeness of links." pith.science (2026). https://pith.science/paper/DN6RDIY3

@misc{pith2026250909615,
  author       = {Pith},
  title        = {Pith review of: Monodromies of surfaces in 3-manifolds, right-veeringness, and primeness of links},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DN6RDIY3}},
  note         = {Machine review of arXiv:2509.09615}
}
abstract

Extending the notion of monodromies associated with open books of $3$-manifolds, we consider monodromies for all incompressible surfaces in $3$-manifolds as partial self-maps of the arc set of the surfaces. We use them to develop a primeness criterion for incompressible surfaces constructed as iterative Murasugi sums in irreducible $3$-manifolds. We also consider a suitable notion of right-veeringness for monodromies of incompressible surfaces. We show strongly quasipositive surfaces are right-veering, thereby generalizing the corresponding result for open books and providing a proof that does not draw on contact geometry. In fact, we characterize when all elements of a family of incompressible surfaces that is closed under positive stabilization are right-veering. The latter also offers a new perspective on the characterization of tight contact structures via right-veeringness as first established by Honda, Kazez, and Mati\'c. As an application to links in $S^3$, we prove visual primeness of a large class of links, the so-called alternative links. This subsumes all prior visual primeness results related to Cromwell's conjecture. The application is enabled by the fact that all links in $S^3$ arise as the boundary of incompressible surfaces, whereas classical open book theory is restricted to fibered links -- those links that arise as the boundary of the page of an open book.

Figures

Figures reproduced from arXiv: 2509.09615 by the authors.

Figure 1
Figure 1. A positive Hopf band H and a product disk D, with ∂0D in blue and ∂1D in red. which ∂0D and ∂1D form cocores of H. In particular, we see that the monodromy ϕ is defined for all arcs and is in fact induced (via an orientation preserving parametrization H ∼= S 1 × [0, 1]) by the positive Dehn twist T : S 1 × [0, 1] → S 1 × [0, 1],(z, t) 7→ (e 2πitz, t). Example 2.4 (Fiber surfaces). Let Σ be a compact surface with non… view at source ↗
Figure 2
Figure 2. for an illustration. The pair (M, Σ) is called the Murasugi sum of (M1, Σ1) and (M2, Σ2) along the summing region P. For simplicity, we also say that Σ is the Murasugi sum of Σ1 and Σ2, suppressing the 3-manifolds in the notation. Moreover, if for j = 1, . . . , n, f1(s2j ) ⊂ Σ1 and f2(s2j−1) ⊂ Σ2 are essential arcs, we say the Murasugi sum (or the summing region) are essential. In another abuse of notation, we deno… view at source ↗
Figure 3
Figure 3. A schematic picture of the disk D, together with its intersec￾tions with M◦ 1 and M◦ 2 , where 1 (green) indicates a 1-disk and 2 (yellow) indicates a 2-disk. In this case, the arc δ is obtained as the union of the negative parts of the boundaries of the 2-disks and the positive part of the boundaries of the 1-disks (note that these coincide on P ′ ). be taken as a guiding example through the rest of the proof. We s… view at source ↗
Figures from the paper (26 more)
Figure 4
Figure 4. Figure 4: The disk D near P ′ ∩ γ, divided into its intersection with M◦ 2 (labeled 2), and M◦ 1 (labeled 1). Here the intersection of D and P is contained in 2. pushes γ ′ ∩ P into M1; see [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: The disk D near P ′ ∩ γ ′ , divided into its intersection with M◦ 1 (labeled 1), and M◦ 2 (labeled 2). Here the intersection of D and P is contained in 1. Claim 3.3. Let I be a connected component of P ′ ∩ D. If both endpoints of I lie on γ, then I union the subinterva…
Figure 6
Figure 6. Figure 6: The bigon formed by a segment s of P ′ , sent to Σ1, and the boundary of P, in case (2). Note that, in particular, the points a and b are on the same side of P, and on the other side of s there is a 2-disk which is a bigon whose boundary is the union of s and a segment…
Figure 7
Figure 7. Figure 7: Using the bigon in P ′ from before (shaded dark red), we can isotope away from P ′ so that we get a disk contained in Σ2. But then we must have another bigon formed by a segment of γ and the boundary of P (shaded blue) which gives the desired contradiction. With the an…
Figure 8
Figure 8. Figure 8: A schematic picture of a disk Di ⊂ M2 [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: A schematic picture of a disk Di ⊂ M1. identification of Σ1 and Σ′ 1 , this implies that Di can be identified with a product disk in M◦ 1 that witnesses δi going to γ ′ i . See [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: The segments δi overlap over P ′ , and moreover join up to a connected arc. This also happens in the bubbles, where the segments δn and δm coincide with a segment of δi. Dj ⊂ M◦ 1 and Dk ⊂ M1; see right-hand side of [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: On the top, an arc γ together with two disks D2, D5 ⊂ M2, going from γ to δ. On the bottom, the disks D1, D3, D4 ⊂ M1 going from δ to γ ′ . Gluing all of the disks together yields the product disk D witnessing ϕ(γ) = γ ′ [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: A disk Di ⊂ M2 that intersects both γ and γ ′ , glued to disks Dj , Dk ⊂ M1. Both of these disks also intersect γ and γ ′ [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]
Figure 13
Figure 13. Figure 13: A disk that intersects γ but not γ ′ [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]
Figure 14
Figure 14. Figure 14: Example of a plumbing region P (red), and arcs γ1, . . . , γ7 starting at the point p ∈ ∂Σ such that [γ1] < [γ2] < [γ3] < [γ4] < [γ5] < [γ6] < [γ7] and [γ1] <P [γ2] =P [γ3] <P [γ4] <P [γ5] <P [γ6] <P [γ7]. such that γ and δ intersect P minimally, and sγ and sδ denote …
Figure 15
Figure 15. Figure 15: Arcs α (black) and α ′ (blue) that represent an isotopy class a = [α] and its image ϕ(a) = [α ′ ], respectively, for an a that satisfies (2). This is, of course, an ‘if and only if’. In other words, for a family P as above, containing at least one surface that is not …
Figure 16
Figure 16. Figure 16: Removing x if R contains part of ∂Σ. Right: the arc β (red). Right: αe′ (blue) is a representative of ϕe(a) with minimal intersection with α. Note that any other intersection points that were between p and x running along α ′ are have no corresponding intersections be…
Figure 17
Figure 17. Figure 17: Removing x if x comes before y on α ′ [PITH_FULL_IMAGE:figures/full_fig_p025_17.png]
Figure 18
Figure 18. Figure 18: Removing x if y comes before x on α ′ [PITH_FULL_IMAGE:figures/full_fig_p025_18.png]
Figure 19
Figure 19. Figure 19: The cases where a becomes right-veering if we perform the previous plumbings. The rectangle whose vertices are x, y, q, w could contain more intersection points. Here is where we refine our choice of x and y: if there is any negative intersection point x ′ that occurs…
Figure 20
Figure 20. Figure 20: The points x and y are chosen to be the ones that feature the largest of the nested rectangles. The case where x occurs before y on α ′ is depicted, for the other case, exchange the labels of the two points and the orientation of the segment of α delimited by x and y.…
Figure 21
Figure 21. Figure 21: Left: the segment αz0 when z1 is not in the interior of α ′ x,y. Right: the segment αz0 when z1 is in the interior of α ′ x,y. As in [PITH_FULL_IMAGE:figures/full_fig_p027_21.png]
Figure 22
Figure 22. Figure 22: On the left, the arc β, which, after intersecting x, comes out in a region with a vertex z that comes before x and y along α ′ . On the right, the green bigon exhibits that all intersection points up to and including x are removed. Then, the purple segment adds every …
Figure 23
Figure 23. Figure 23: Left, the plumbing arc β. Right, all the intersection points before x and y get removed, and only the ones between z0 (excluded) and z1 (included and labeled z ′ 1 in αe′) get added back (in the figure only z ′ 1 is shown, but there could be more), and with the same s…
Figure 24
Figure 24. Figure 24: Doing two plumbings when all sides of S are between x and y to reduce to one of the previous cases (at the expense of an extra positive intersection point near p). Note that the second plumbing is also one of the previous cases, as the region R ′ has a side that comes…
Figure 25
Figure 25. Figure 25: for an illustration. Note that since y1 is not removed, in particular α stays left-veering at q [PITH_FULL_IMAGE:figures/full_fig_p029_25.png]
Figure 26
Figure 26. Figure 26: The fact that β ∩ Σg,1 is essential ensures that a is still left-veering for the new monodromy ϕe, and the fact that it is nonseparating ensures that the new region R contains the boundary, so we have reduced to the previous case and we are done. No intersections. The…
Figure 26
Figure 26. Figure 26: The case where there is a unique (positive) intersection point, and R has one puncture and does not contain parts of the boundary of ∂Σ. The two black circles represent one of the genera in R, so that β ∩ R is non-separating. On the top right, we perform the positive …
Figure 27
Figure 27. Figure 27: The surface Σ and the annuli A and AD, for minimally intersecting representatives γ and γ ′ of a and ϕ(a). being parallel away from a neighborhood of the start and endpoint of a, up to a correction of ± 1 2 at the start and endpoint. Since a < ϕ(a) and a > ϕ ¯ (a¯) (o…
Figure 28
Figure 28. Figure 28: On the left, we have that the arc b ′ is fixed in Σv because it is contained in D. On the right, b ′ is obtained from b by an isotopy on the boundary, so it is also fixed in Σv′ . Note that b ′ could be boundary parallel in Σv, but not in Σv′ as there must be—at least…

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    write newline

    " write newline "" before.all 'output.state := FUNCTION output.nonempty.mrnumber duplicate missing pop "" 'skip if duplicate empty 'pop " " swap * " " * write if FUNCTION fin.entry add.period write newline INTEGERS nameptr namesleft numnames FUNCTION format.language language e...

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.