Ordering curves on surfaces

Binbin Xu; Hanh Vo; Hugo Parlier

REVIEW 2 major objections 5 minor 1 cited by

Ordering curves on surfaces

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that the complete ordering of closed geodesic lengths by size is an injective invariant: no two distinct hyperbolic metrics on the same topological surface rank all interior curves identically.

desk verdict The pants projective injection and the never-changing-order theorem are real additions, but the proof of the main theorem misses Fenchel-Nielsen twists on general surfaces with boundary, so the big claim is not established as written. read the letter →

arxiv 2506.06481 v1 pith:PLPL3DTD submitted 2025-06-06 math.GT

Hugo Parlier , Hanh Vo , Binbin Xu This is my paper

classification math.GT MSC 57K2057M5032G1537F30

keywords Teichmüllerspaceslengthordersk-systoleshyperbolicsurfacesclosedgeodesicspairsofpantsprojectiveinjectivityself-intersectionnumbers

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 proves that on any finite-type orientable hyperbolic surface with negative Euler characteristic, the complete order of lengths of all interior closed geodesics—not the numerical lengths, only which curve is shorter than which—determines the hyperbolic metric up to isotopy. Two distinct metrics on the same topological surface must disagree on the relative ranking of at least two interior curves. The proof first shows that finitely many length ratios already separate metrics: for every such surface there is a finite list of interior curves whose length ratios pin down the metric. In the opposite direction, the paper finds families of word-pairs on a pair of pants whose length inequality is valid for every hyperbolic metric, and uses this to show that for $k = 1, 2, 3, 4$ the shortest curve with at least $k$ self-intersections on any hyperbolic pair of pants self-intersects exactly $k$ times.

What carries the argument

The load-bearing mechanism is a projectively injective length map $\Phi : T(\Sigma) \to \mathbb{R}^r$, $X \mapsto (\ell_X([\gamma_1]), \dots, \ell_X([\gamma_r]))$, meaning proportional output forces equal metrics. On a pair of pants the construction uses Fricke–Vogt trace coordinates for $\mathrm{SL}(2,\mathbb{R})$ representations of the free group on two generators: a representation is determined up to conjugacy by the four traces $(\operatorname{tr}\rho(u), \operatorname{tr}\rho(v), \operatorname{tr}\rho(uv), \operatorname{tr}\rho(u^{-1}v))$, so four interior curves realizing those conjugacy classes give a map into $\mathbb{R}^4$ with the required property. A second mechanism, used for Theorem 1.3, is the decomposition of a hyperbolic pair of pants into two isometric geodesic hexagons by three pairwise orthogonal geodesic arcs; the reflection involution exchanging the two hexagons lets the authors replace long subpaths of a concatenated geodesic by shorter reflected subpaths, turning the concatenation inequality into a strict length inequality valid for every metric.

What would settle it

A concrete way to test the main claim is to search for two distinct hyperbolic metrics on the same surface whose length ratios agree on the finite curve list constructed in Theorem 1.2; if such a pair existed, the projective injectivity used in the proof would fail, and Theorem 1.1 would be false, since two distinct metrics with identical ratios on all the selected curves would still have to be separated by some order comparison in the proof.

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Extended reading notes

Core claim

The central claim, Theorem 1.1, is that the map sending a hyperbolic metric $X$ on a fixed surface $\Sigma$ to the total order of the lengths $\ell_X([\gamma])$ over all interior curve homotopy classes $[\gamma]$ is injective on Teichmüller space $T(\Sigma)$. The engine is Theorem 1.2, which produces a finite collection of interior curves $\gamma_1, \dots, \gamma_r$ with a projectively injective length map: if the length vectors of two metrics are proportional, the metrics coincide. For a pair of pants, four interior curves suffice, via the Fricke–Vogt trace-coordinate description of the character variety of the free group on two generators; the four traces $(\operatorname{tr}\rho(u), \operatorname{tr}\rho(v), \operatorname{tr}\rho(uv), \operatorname{tr}\rho(u^{-1}v))$ determine the representation up to conjugacy, and hence the metric. The paper completes the earlier simple-curve result of McShane–Parlier by allowing all curves and removing restrictions on boundary lengths, and it proves Theorem 1.3, a metric-independent inequality for certain concatenated words, leading to Corollary 1.4 on $k$-systoles of pairs of pants for $k \le 4$.

Load-bearing premise

The load-bearing premise is that a finite cover of the surface by embedded pairs of pants, each carrying a projectively injective four-curve length map, combines into a projectively injective map for the whole surface; the paper asserts this gluing in a single sentence, without a detailed verification of how the inter-pants gluing data are recovered by the chosen curves.

Editorial extensions

If this is right

If Theorem 1.1 is correct, the length order is a complete invariant: two hyperbolic metrics on the same surface coincide exactly when every pair of interior curves has the same relative order of lengths.
Theorem 1.2 gives a finite certificate for metric equality: on every such surface a finite list of interior curves separates all metrics by length ratios alone.
Theorem 1.3 supplies universal length comparisons that hold without knowing the metric, so they can prune candidate curves in searches for shortest curves.
Corollary 1.4 confirms, for $k = 1, 2, 3, 4$, that the $k$-systole of any hyperbolic pair of pants self-intersects exactly $k$ times.
For larger $k$, the method reduces the $k$-systole search to comparing finitely many explicit candidate words, though the candidate lists grow quickly.

Reading between the lines

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

Editorial inference: since the length order is a total order on a countable set, injectivity of the order map suggests Teichmüller space embeds into a space of countable total orders, giving a purely combinatorial coordinatization that the paper does not explicitly state.
Editorial inference: the finite curve set from Theorem 1.2 could be used as a computational certificate—two metrics are equal if and only if their length ratios agree on that finite list—which would make the abstract injectivity result algorithmically checkable.
Editorial inference: the reflection-involution argument on pairs of pants may generalize to other surfaces with an orientation-reversing involution, producing further metric-independent word inequalities beyond the pairs covered by Theorem 1.3.
Editorial inference: the $k$-systole computation could in principle be extended beyond $k = 4$ by automating the comparisons of the longer candidate lists with the same word-length and self-intersection techniques, although the paper does not claim this.

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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

Main thing you should know: the paper has a good idea and one genuinely new construction, but the advertised Theorem 1.1 is not proved as written for surfaces with interior cuffs. The stress-test note is right. In the proof of Theorem 1.2 for general surfaces, the authors cover the surface by embedded pants, apply Corollary 3.8 to each, and assert that metric information on the pants determines the metric on the whole surface. That is only true if you know the restrictions together with the gluing data. From lengths alone you get cuff lengths, but not the Fenchel-Nielsen twists along interior cuffs. The curves chosen in that paragraph all live inside a single pants or are the cuff curves themselves, so their lengths are unchanged when you twist along an interior cuff. The finite collection is constant along a positive-dimensional twist family and cannot be projectively injective. For closed surfaces the proof does add transversal curves; for surfaces with boundary it does not. So Theorem 1.2 is unproved for the general finite-type case, and Theorem 1.1 currently rests on that.

That said, there is real substance here. The four-curve projective injection for a pair of pants, Corollary 3.8, is new and the character-variety argument is coherent. Theorem 1.3 is also interesting and appears to hold; the k-systole computation for k up to 4 is concrete and the code is available. The citation pattern is fine; leaning on [66] for technical lemmas is legitimate because the new results genuinely extend that work.

Soft spots in proportion: the twist gap is load-bearing, not cosmetic. Proposition 3.1 has a wording slip where the proof assumes hyperbolicity of M2 and N2 even though the proposition does not; that one is minor because the trace product is forced nonzero. The proof of Proposition 3.7 is terse but the case analysis is plausible. The self-intersection bounds for the pants cases are stated without detailed derivations, but for the concrete curves listed they look believable. The k = 4 smoothing inequalities are asserted more than proved, though the figures help.

This paper is for Teichmüller theory readers. Theorem 1.3 and the pants injection deserve attention. The main theorem needs a repaired proof, probably by adding curves transverse to interior cuffs or by a different global argument. I would send this to a serious referee rather than desk reject, with a clear request to focus on the general-surface case in Theorem 1.2.

Referee Report

2 major / 5 minor

Summary. The paper studies the order of lengths of closed geodesics on hyperbolic surfaces of finite type with negative Euler characteristic. Its main result (Theorem 1.1) states that the map sending a hyperbolic metric to the total order of lengths of all interior closed geodesics is injective. The proof proceeds by first constructing, for each surface, a finite collection of interior curves whose length vector is projectively injective (Theorem 1.2), then converting a difference in a length ratio into a reversal of order of two concatenated curves. The paper also proves (Theorem 1.3) that for any pair of pants, if u and v are cyclically reduced words with u starting and ending in different letters, then ℓ(u) < ℓ(uv) for every hyperbolic metric, and uses this to show (Corollary 1.4) that for 1 ≤ k ≤ 4 the shortest geodesic with at least k self-intersections has exactly k self-intersections. The paper includes a computational verification of the candidate lists for small k with code available.

Significance. If Theorem 1.1 is correct, it is a significant rigidity result: the complete length order of all interior closed geodesics is a total invariant of the hyperbolic metric, extending prior work of McShane and Parlier for simple curves on closed surfaces and addressing the remaining cases of pairs of pants and surfaces with boundary. The proof strategy, based on Fricke–Klein trace identities and projective injections into Euclidean space, is natural and elegant. The k-systole corollary is a nice application, and the authors provide machine-checkable code for the finite enumeration, which is a strength. However, as detailed below, the proof of Theorem 1.2 for surfaces with boundary has an unaddressed gluing issue that currently leaves Theorem 1.1 unsupported in the general case, and the k-systole argument relies on an unproved word-length assertion.

major comments (2)

[Section 3, proof of Theorem 1.2 (paragraph beginning 'For a surface Σ_{g,n} with more complicated topology...')] The proof asserts that 'the metric information on these pairs of pants will determine the metric information on Σ_{g,n}' and then applies Corollary 3.8 pants-by-pants. This is not justified: the four curves chosen for each pair of pants (a, b, ab, ab^{-1} or a, a^{-1}b, a^{-2}b, b) lie in a single pants or are the cuff curves themselves, so their lengths are unchanged by Fenchel–Nielsen twists along the interior cuffs of the pants decomposition. Consequently, the finite collection Φ built in this paragraph is constant along positive-dimensional twist families and cannot be projectively injective. Since the proof of Theorem 1.1 in Section 4 invokes Theorem 1.2 to obtain two curves with different length ratios, Theorem 1.1 is not proved for surfaces with boundary that require a pants decomposition with interior cuffs (e.g., Σ_{0,4} or Σ_{1,2}). A repair requires adding curves that cross each interior cuff, or an explicit open-cover argument in which the overlap identifications are determined by the chosen lengths.
[Section 5.3, paragraph before Corollary 5.5] The statement 'For small values of k, to have k self-intersections, the curve needs at least 2(k+1) letters in its cyclically reduced and admissible word' is asserted without proof or reference. This claim is load-bearing for the reduction to Baribaud's theorem and for the exhaustiveness of the candidate list S≥k: if a k-self-intersecting geodesic could have fewer than 2(k+1) letters, Baribaud's lower bound would not apply to it and the comparison of lengths might miss the true k-systole. The authors should either prove this assertion for k = 1, 2, 3, 4 or state precisely that it is verified by the exhaustive computation, explaining the search bound used by the code.

minor comments (5)

[Lemma 3.5] The displayed simplification of the derivative of cosh(tx)/cosh(ty) is algebraically incorrect; the correct numerator is ((x−y)/2)sinh(t(x+y)) + ((x+y)/2)sinh(t(x−y)), which is still positive for x > y ≥ 0, so the lemma's conclusion is unaffected.
[Throughout] There are numerous cross-reference artifacts such as 'Theorem 1.31.3', 'Corollary 1.41.4', 'Theorem 1.21.2', and 'Remark 2.32.3' that should be fixed in the final version.
[Section 5.3, algorithm paragraph] The URL for the code is duplicated ('https://github.com/hanhv/small-k-systoleshttps://github.com/hanhv/small-k-systoles') and should be cleaned up.
[Section 5.3, algorithm paragraph] The phrase 'aBk' appears to be a typo for 'ab^{-k}' or similar; please correct the notation.
[Section 3, proof of Theorem 1.2 (closed-surface case)] The sentence 'In each pair of pants decomposition of Σg,0, there a non-separating curve' is missing the verb 'is' and should read 'there is a non-separating curve'.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the main theorems are proved from trace identities, geometric length comparisons, and independent prior results; the general-surface gluing step has a real twist-recovery gap, but that is a correctness issue rather than a self-referential reduction.

full rationale

The derivation chain is not circular. Theorem 1.1 is obtained from Theorem 1.2 by a limit argument comparing two length ratios, and Theorem 1.2 is obtained for pants from Proposition 3.4, whose proof uses the Fricke-Vogt trace theorem, Lemma 3.5, and Lemma 3.6 borrowed from [66]. The cited items from [66] are independent published results with stated assumptions, so even though the citation overlaps with one of the present authors, the paper is not reducing its claim to an unverified self-citation. The k-systole conclusion uses Baribaud's theorem and the Despré-Lazarus algorithm, both external to this paper. The only flagged issue is in the proof of Theorem 1.2 for general surfaces, where the sentence 'Notice that the metric information on these pairs of pants will determine the metric information on Σg,n' asserts without proof that gluing twists are recovered; indeed the four curves chosen inside each pants do not detect Fenchel-Nielsen twist parameters. This is a possible correctness gap in the argument, not a case of a conclusion being equivalent to its inputs by construction, so it does not raise the circularity score. The score of 2 reflects only the presence of self-citations that are legitimate but not independently re-proved here.

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

The central results rest on standard character variety theory (Fricke-Vogt), a cited lemma from prior work, and two unproved structural assumptions: the orthogeodesic hexagon setup for pants, and the gluing of projectively injective maps across a pants decomposition. The k-systole argument adds an unproved combinatorial word-length bound. No free parameters are fitted, and no new entities are invented.

assumptions (5)

standard math Fricke-Vogt theorem: a conjugation-invariant regular function of two SL(2,C) matrices is a polynomial in (tr M, tr N, tr MN), and an irreducible representation is determined up to conjugacy by this triple.
Used in Proposition 3.1 to conclude that matching traces imply conjugacy.
standard math Lemma 3.6 (from McShane-Parlier [66]): a function of the form cosh(x1 t)+cosh(x2 t)-sum_k cosh(y_k t) with x1 > y_k has at most one positive zero.
Borrowed without proof from [66]; it is load-bearing in Proposition 3.7 for the uniqueness of t=1.
domain assumption For any hyperbolic metric on a pair of pants, the three common perpendicular geodesic arcs between boundary geodesics realize the combinatorial hexagon cutting, and reflection in these arcs is an isometry exchanging the two hexagons.
Used throughout Section 5 to translate between geodesic words and piecewise geodesic comparisons (e.g., Propositions 5.2 and 5.3). This is standard in hyperbolic geometry but is an unstated setup.
domain assumption Metric information on a finite collection of pairs of pants covering a surface determines the metric on the whole surface, and projectively injective maps on the pants combine into a projectively injective map on the Teichmüller space.
Used in the proof of Theorem 1.2 for general surfaces; stated without proof.
ad hoc to paper For k ≤ 4, a geodesic on a pair of pants with k self-intersections needs at least 2(k+1) letters in its cyclically reduced admissible word.
Used in Section 5.3 to reduce the k-systole search to candidates with at most 2k strings; no proof is given in the paper, only asserted for small k.

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Cite this review

Pith. "Pith review of Ordering curves on surfaces." pith.science (2026). https://pith.science/paper/PLPL3DTD

@misc{pith2026250606481,
  author       = {Pith},
  title        = {Pith review of: Ordering curves on surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PLPL3DTD}},
  note         = {Machine review of arXiv:2506.06481}
}

read the original abstract

We study the order of lengths of closed geodesics on hyperbolic surfaces. Our first main result is that the order of lengths of curves determine a point in Teichm\"uller space. In an opposite direction, we identify classes of curves whose order never changes, independently of the choice of hyperbolic metric. We use this result to identify short curves with small intersections on pairs of pants.

Figures

Figures reproduced from arXiv: 2506.06481 by the authors.

**Figure 2.** The plus and minus versions of ξi Let w = t11 · · · t1n1 t21 · · · t2n2 · · · tk1 · · · tknk be a word of letters in {ξ ± 1 , ξ ± 2 , ξ ± 3 }, such that for any j ∈ {1, ..., k}, all letters tj1, ..., tjnj are in {ξ + 1 , ξ + 2 , ξ + 3 } or {ξ − 1 , ξ − 2 , ξ − 3 } at the same time. Since both H+ and H− are topological disks, the word w ′ = t11t21 · · · tk1 induces a curve homotopic to one induced by w. A cyclic perm… view at source ↗

**Figure 3.** Visualizing the homotopies ble words. There is thus a one-to-one map from the set of homotopy classes of oriented curves on P to W, which sends [η] to [w(η)]. When P is equipped a hyperbolic metric, the above discussion can be realized geometrically. Let X be a hyperbolic metric on P with geodesic boundary (it is convenient to think of all three cuffs are realized by non-zero length simple closed geodesics, although… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Reducing lengths of geodesics via the involution of a pair of pants, I [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]

**Figure 5.** Figure 5: Reducing lengths of geodesics via the involution of a pair of pants, II [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]

**Figure 6.** Figure 6: ℓX(aaba−1 b) > ℓX(aaba−1 b −1 ) 25 [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]

**Figure 7.** Figure 7: ℓX(abbab−1 ) > ℓX(aaba−1 b −1 ) abab−1 a −1 b −1 abab−1 a −1 b a b a b [PITH_FULL_IMAGE:figures/full_fig_p026_7.png]

**Figure 8.** Figure 8: ℓX(abab−1 a −1 b −1 ) > ℓX(abab−1 a −1 b) Remark 5.6. One should notice that, when k is large, we can produce k self-intersection number with k ′ geodesic segments connecting ξ1, ξ2 and ξ3 in P, for k ′ < 2(k + 1). By results of the second author [7], there are example…

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Oriented graphs on curve complex I: hyperbolic and extremal length
math.GT 2025-07 conditional novelty 8.0 of 10

The comparison graph of lengths of disjoint simple closed curves determines a closed hyperbolic surface, and every automorphism of this graph is geometric.

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages · cited by 1 Pith paper

[1]

Baribaud

Claire M. Baribaud. Closed geodesics on pairs of pants. Israel J. Math., 109:339–347, 1999

work page 1999
[2]

Relations between word length, hyperbolic length and self-intersection number of curves on surfaces

Moira Chas. Relations between word length, hyperbolic length and self-intersection number of curves on surfaces. In Recent advances in mathematics, volume 21 of Ramanujan Math. Soc. Lect. Notes Ser., pages 45–75. Ramanujan Math. Soc., Mysore, 2015

work page 2015
[3]

Computing the geometric intersection number of curves

Vincent Despré and Francis Lazarus. Computing the geometric intersection number of curves. J. ACM, 66(6):Art. 45, 49, 2019. 26

work page 2019
[4]

Notiz über die Discontinuität gewisser Collineationsgruppen

Robert Fricke. Notiz über die Discontinuität gewisser Collineationsgruppen. Math. Ann., 47(4):557–563, 1896

work page
[5]

William M. Goldman. Trace coordinates on Fricke spaces of some simple hyperbolic surfaces. In Handbook of Teichmüller theory. Vol. II, volume 13 of IRMA Lect. Math. Theor. Phys., pages 611–684. Eur. Math. Soc., Zürich, 2009

work page 2009
[6]

Multiplicities of simple closed geodesics and hypersur- faces in Teichmüller space

Greg McShane and Hugo Parlier. Multiplicities of simple closed geodesics and hypersur- faces in Teichmüller space. Geom. Topol., 12(4):1883–1919, 2008

work page 1919
[7]

Short closed geodesics on cusped hyperbolic surfaces.Pacific J

Hanh Vo. Short closed geodesics on cusped hyperbolic surfaces.Pacific J. Math., 318(1):127– 151, 2022

work page 2022
[8]

H. Vogt. Sur les invariants fondamentaux des équations différentielles linéaires du second ordre. Ann. Sci. École Norm. Sup. (3), 6:3–71, 1889. Addresses: Department of Mathematics, University of Fribourg, Fribourg, Switzerland Department of Mathematics, Purdue University, Indiana, USA School of Mathematical Sciences, Nankai University, Tianjin, China Ema...

work page

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Baribaud

Claire M. Baribaud. Closed geodesics on pairs of pants. Israel J. Math., 109:339–347, 1999

work page 1999

[2] [2]

Relations between word length, hyperbolic length and self-intersection number of curves on surfaces

Moira Chas. Relations between word length, hyperbolic length and self-intersection number of curves on surfaces. In Recent advances in mathematics, volume 21 of Ramanujan Math. Soc. Lect. Notes Ser., pages 45–75. Ramanujan Math. Soc., Mysore, 2015

work page 2015

[3] [3]

Computing the geometric intersection number of curves

Vincent Despré and Francis Lazarus. Computing the geometric intersection number of curves. J. ACM, 66(6):Art. 45, 49, 2019. 26

work page 2019

[4] [4]

Notiz über die Discontinuität gewisser Collineationsgruppen

Robert Fricke. Notiz über die Discontinuität gewisser Collineationsgruppen. Math. Ann., 47(4):557–563, 1896

work page

[5] [5]

William M. Goldman. Trace coordinates on Fricke spaces of some simple hyperbolic surfaces. In Handbook of Teichmüller theory. Vol. II, volume 13 of IRMA Lect. Math. Theor. Phys., pages 611–684. Eur. Math. Soc., Zürich, 2009

work page 2009

[6] [6]

Multiplicities of simple closed geodesics and hypersur- faces in Teichmüller space

Greg McShane and Hugo Parlier. Multiplicities of simple closed geodesics and hypersur- faces in Teichmüller space. Geom. Topol., 12(4):1883–1919, 2008

work page 1919

[7] [7]

Short closed geodesics on cusped hyperbolic surfaces.Pacific J

Hanh Vo. Short closed geodesics on cusped hyperbolic surfaces.Pacific J. Math., 318(1):127– 151, 2022

work page 2022

[8] [8]

H. Vogt. Sur les invariants fondamentaux des équations différentielles linéaires du second ordre. Ann. Sci. École Norm. Sup. (3), 6:3–71, 1889. Addresses: Department of Mathematics, University of Fribourg, Fribourg, Switzerland Department of Mathematics, Purdue University, Indiana, USA School of Mathematical Sciences, Nankai University, Tianjin, China Ema...

work page