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arxiv: 2504.17316 · v3 · submitted 2025-04-24 · 🧮 math.GT · math.MG

Small genus, small index critical points of the systole function

Ni An , Ferdinand Ihringer , Ingrid Irmer This is my paper

Pith reviewed 2026-05-22 19:12 UTC · model grok-4.3

classification 🧮 math.GT math.MG
keywords systole functionTeichmüller spaceThurston spineminimal filling setscritical pointshyperbolic surfacesgenus 5integer linear programming
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The pith

Critical points of the systole function in genus 5 produce strata in the Thurston spine where systoles do not basis the surface homology.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes the index of a family of critical points of the systole function on Teichmüller space. These points had previously only been found in surfaces of very large genus, making their index calculations infeasible. The new examples in genus 5 show that the shortest closed geodesics at the critical point can fail to span the full homology of the surface. The authors support this by identifying an explicit minimal filling set of eight systoles and by using integer linear programming to bound the possible sizes of such sets for certain regular right-angled polygon tessellations.

Core claim

The authors establish a family of critical points of the systole function whose indices can now be calculated because the points occur in small genus; in genus 5 this family yields a minimal filling set of cardinality 8 that is presumably realized by systoles, and more generally they determine the smallest and largest possible cardinalities of minimal filling sets arising from tessellations by regular right-angled m-gons for m equal to 5, 6 or 7.

What carries the argument

Minimal filling subsets of the systoles at each critical point, found via integer linear programming on tessellations by regular right-angled m-gons together with symmetry breaking.

If this is right

  • Strata appear in the Thurston spine for which the systoles fail to determine a basis for the homology of the surface.
  • Genus 5 admits a minimal filling set of eight simple closed geodesics realized as systoles at a critical point.
  • For tessellations by regular right-angled pentagons, hexagons or heptagons the possible cardinalities of minimal filling sets are bounded above and below.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same computational search could be rerun in genus 4 to test whether still smaller examples exist.
  • The explicit cardinality-8 example supplies a concrete test case for algorithms that enumerate short geodesics on low-genus surfaces.
  • The bounds on filling-set size for the three families of tessellations may extend to other regular tilings that appear in the study of hyperbolic 3-manifolds.

Load-bearing premise

The critical points located by the computation are actually attained by systoles on some hyperbolic surface and the integer-linear-programming solutions correspond to geometrically realizable collections of curves.

What would settle it

An explicit hyperbolic metric on a genus-5 surface whose shortest geodesics form a homology basis or whose minimal filling set has cardinality other than 8 would contradict the claimed family of critical points.

Figures

Figures reproduced from arXiv: 2504.17316 by Ferdinand Ihringer, Ingrid Irmer, Ni An.

Figure 1
Figure 1. Figure 1: Examples with automorphism groups given by quotients of triangle groups have been intensely studied, for example [5, 24]. This is partly because as shown in Theorem 37 of [23], the large symmetry groups force such examples to be critical points of all mapping class group-equivariant Morse function on Tg [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Four right angled hexagons (not drawn to scale, with alternating purple and blue edges) of side length s in the tesselation. The dual C 6 is shown in grey, and a couple of corner arcs aj and aj−1 are shown in red. least cardinality 3 in order to give a curve with combinatorial length at least 4, this gives the required contradiction to the existence of c ′ . □ 2.1. Parameterising the systoles. This subsect… view at source ↗
Figure 3
Figure 3. Figure 3: The loop in green is the boundary of a square in the 4-cube that is mapped to a homotopically nontrivial loop on the torus, whereas the loop in purple is the boundary of a square in the 4-cube that is mapped to a square in the torus. Proof. This is a proof by induction. The lemma holds in the case m = 3. This is a decom￾position of the 2-sphere into 8 right-angled triangles, with dual graph the 1-skeleton … view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: All systoles in the genus 5 example [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A minimal filling set of 8 geodesics in the genus 5 surface [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: A fundamental domain for C 5 in the disk model. Edges that correspond to the same number are glued together in the surface. of the jth pair of edges, where the jth pair contains the ith edge. This gives the constraint (5) Mn . . . M1 = I in P SL(2, R) These constraints make it possible to eliminate dependent angle and length parame￾ters. For example, if our minimal filling set C ′ has exactly M(m) intersec… view at source ↗
Figure 8
Figure 8. Figure 8: Mj is a composition of scaling and rotation. to calculate the length of the geodesic intersecting the same edges of F(C ′ ) in the same order as c. For any c ∈ C, we can numerically compute ∂L(c) ∂yi and ∂L(c) ∂θj for some independent param￾eters yi and θj . The number of linearly independent differentials {dL(c)(x) | c ∈ C} then gives the index of the critical point x. 3.3. Questions and future work. The … view at source ↗
read the original abstract

In this paper the index of a family of critical points of the systole function on Teichm\"uller space is calculated. The members of this family are interesting in that their existence implies the existence of strata in the Thurston spine for which the systoles do not determine a basis for the homology of the surface. Previously, index calculations of critical points with this pathological feature were impossible, because the only known examples were in surfaces with huge genus. A related concept is that of a ``minimal filling subset'' of the systoles at the critical point. Such minimal filling sets are studied, as they relate to the dimension of the Thurston spine near the critical point. We find an example of a minimal filling set of simple closed geodesics in genus 5 with cardinality 8, that are presumably realised as systoles. More generally, we determine the smallest and largest cardinality of a minimal filling set related to a tesselation of a hyperbolic surface by regular, right-angled $m$-gons for $m \in \{ 5, 6, 7 \}$. For this, we use integer linear programming together with a hand-tailored symmetry breaking technique.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper computes the indices of a family of critical points of the systole function on Teichmüller space of hyperbolic surfaces. These critical points are associated with minimal filling subsets of systoles that fail to span the homology of the surface, implying the existence of corresponding strata in the Thurston spine. The authors provide an explicit example of a minimal filling set of cardinality 8 in genus 5 (presumably realized by systoles) and, more generally, determine the minimal and maximal cardinalities of such sets for regular right-angled m-gon tessellations with m=5,6,7 via integer linear programming combined with a hand-tailored symmetry-breaking technique.

Significance. If the geometric realizations of the ILP solutions as actual systoles at critical points can be confirmed, the work supplies the first small-genus examples of such pathological critical points, making index calculations feasible where only high-genus examples were previously known. The computational determination of filling-set cardinalities via ILP offers a concrete method for probing the local dimension of the Thurston spine near these points.

major comments (2)
  1. [Abstract and computational results section] Abstract and computational results section: The central claims (index calculations and the implication for Thurston-spine strata) rest on the assumption that the ILP-derived minimal filling sets are realized by systoles at a critical point. No explicit verification is provided that the selected curves achieve equal minimal length, that no shorter curves exist, or that the configuration is indeed critical in Teichmüller space. This is load-bearing for the index and spine-strata conclusions.
  2. [Computational results section] Computational results section: The integer linear programming solutions for cardinalities (including the genus-5 cardinality-8 example and the extremal values for m=5,6,7) lack accompanying error analysis, bounds on solution quality, or geometric checks confirming realizability on hyperbolic surfaces with the regular right-angled tessellations.
minor comments (2)
  1. [Abstract] The phrase 'presumably realised as systoles' in the abstract should be replaced by a precise statement of what has and has not been verified, with a forward reference to the relevant computational section.
  2. Notation for the tessellations (regular right-angled m-gons) and the precise definition of 'minimal filling set' should be introduced with a short preliminary subsection before the ILP discussion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below, clarifying the scope of our claims regarding assumptions and the exact nature of the computational results. We will incorporate revisions to make these points more explicit.

read point-by-point responses

  1. Referee: [Abstract and computational results section] Abstract and computational results section: The central claims (index calculations and the implication for Thurston-spine strata) rest on the assumption that the ILP-derived minimal filling sets are realized by systoles at a critical point. No explicit verification is provided that the selected curves achieve equal minimal length, that no shorter curves exist, or that the configuration is indeed critical in Teichmüller space. This is load-bearing for the index and spine-strata conclusions.

    Authors: We agree that no explicit geometric verification is provided that the ILP-derived sets are realized by actual systoles of equal length with no shorter curves present, nor that the configuration is critical. The manuscript already qualifies this with the phrase 'presumably realised as systoles' and presents the index calculations and Thurston-spine implications conditionally on the existence of such critical points. The combinatorial index formula itself is derived from the minimal filling set data. We will revise the abstract and computational results section to state the assumption more explicitly and to emphasize that the results are conditional on geometric realization. revision: partial

  2. Referee: [Computational results section] Computational results section: The integer linear programming solutions for cardinalities (including the genus-5 cardinality-8 example and the extremal values for m=5,6,7) lack accompanying error analysis, bounds on solution quality, or geometric checks confirming realizability on hyperbolic surfaces with the regular right-angled tessellations.

    Authors: The ILP computations determine exact minimal and maximal cardinalities for minimal filling sets in the combinatorial setting of the regular right-angled m-gon tessellations (m=5,6,7), using a solver that certifies optimality upon termination together with a hand-tailored symmetry-breaking method that enumerates distinct configurations up to the tessellation's symmetry group. The reported values, including the genus-5 cardinality-8 example, are therefore exact for this discrete problem. We did not perform geometric checks or length-equalization verifications because the focus of the section is the combinatorial bounds that inform possible dimensions of Thurston-spine strata. We will add details on the solver, termination criteria, and optimality confirmation to the computational results section; geometric realizability checks are left for future investigation. revision: partial

Circularity Check

0 steps flagged

No circularity: direct index computations and ILP on explicit tessellations

full rationale

The paper computes indices of critical points of the systole function via direct methods on Teichmüller space and determines minimal filling sets using integer linear programming with symmetry breaking on regular right-angled m-gon tessellations (m=5,6,7). These steps rely on explicit combinatorial enumeration and optimization rather than any derivation that reduces to fitted parameters, self-definitions, or self-citation chains. The implication for Thurston spine strata follows from the computed existence of such points, with geometric realization stated as a presumption rather than a circularly derived claim. No load-bearing step equates outputs to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background results from hyperbolic geometry and Teichmüller theory; no free parameters are introduced and no new entities are postulated.

axioms (2)
  • standard math Standard properties of the systole function and critical points on Teichmüller space
    Invoked throughout the definition of the family of critical points and the Thurston spine strata.
  • standard math Existence and uniqueness properties of hyperbolic structures and regular right-angled polygon tesselations
    Used to set up the integer linear programming instances for minimal filling sets.

pith-pipeline@v0.9.0 · 5738 in / 1408 out tokens · 37581 ms · 2026-05-22T19:12:01.560876+00:00 · methodology

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Reference graph

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