A construction of minimal coherent filling pairs
Pith reviewed 2026-05-24 09:31 UTC · model grok-4.3
The pith
A geometric procedure starting from a torus produces minimal coherent filling pairs on every higher-genus surface.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We give a simple geometric procedure for constructing minimal intersecting coherent filling pairs on S_g, g ≥ 3, from the starting point of a coherent filling pair of curves on a torus. Coherent filling pairs have a natural correspondence to square-tiled surfaces, or origamis, and we discuss the origami obtained from the construction.
What carries the argument
The geometric procedure that modifies a coherent filling pair on the torus to produce one on S_g while preserving the filling property, the equality of geometric and absolute algebraic intersection numbers, and minimality of the intersection number.
If this is right
- Minimal coherent filling pairs exist on every closed orientable surface of genus at least 3.
- Each such pair corresponds to a square-tiled surface or origami obtained directly from the torus example.
- The intersection number of these constructed pairs equals the minimal intersection number among all filling pairs on S_g.
Where Pith is reading between the lines
- Iterating the procedure could produce infinite families of distinct minimal coherent pairs on a fixed surface.
- The construction supplies explicit examples that can be used to test conjectures about the mapping class group action on filling pairs.
- The correspondence to origamis may allow the method to be rephrased in terms of flat metrics or translation surfaces.
Load-bearing premise
That the geometric procedure always preserves filling, coherence, and minimality when it is applied to any coherent filling pair on the torus.
What would settle it
A concrete torus coherent filling pair to which the procedure is applied and the resulting pair on S_3 has geometric intersection number strictly larger than the smallest known intersection number among all filling pairs on the genus-3 surface.
read the original abstract
Let $S_g$ denote the genus $g$ closed orientable surface. A \emph{coherent filling pair} of simple closed curves, $(\alpha,\beta)$ in $S_g$, is a filling pair that has its geometric intersection number equal to the absolute value of its algebraic intersection number. A \emph{minimally intersecting} filling pair, $(\alpha,\beta)$ in $S_g$, is one whose intersection number is the minimal among all filling pairs of $S_g$. In this paper, we give a simple geometric procedure for constructing minimal intersecting coherent filling pairs on $S_g, \ g \geq 3,$ from the starting point of a coherent filling pair of curves on a torus. Coherent filling pairs have a natural correspondence to square-tiled surfaces, or {\em origamis}, and we discuss the origami obtained from the construction.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript claims to provide a simple geometric procedure that starts from a coherent filling pair of curves on the torus (with intersection number 1) and produces coherent filling pairs on the closed surface S_g for each g ≥ 3. These output pairs are asserted to be filling, to satisfy geometric intersection number equal to the absolute value of algebraic intersection number, and to achieve the absolute minimal possible geometric intersection number among all filling pairs on S_g. The paper also discusses the square-tiled surface (origami) associated to each constructed pair.
Significance. If the construction is shown to produce pairs whose intersection number meets the known lower bound for filling pairs on S_g, the result supplies an explicit, geometrically described family of minimal coherent filling pairs and their corresponding origamis. This would be useful for explicit computations and for studying the relationship between filling pairs and square-tiled surfaces.
major comments (2)
- [§4] §4 (Proof that the output pair is minimal): the manuscript shows that the constructed pair has a small intersection number obtained by the geometric lifting procedure, but does not recall or cite the known lower bound on the minimal intersection number of any filling pair on S_g. Without an explicit comparison showing that the output intersection equals this lower bound (rather than merely being smaller than some other pairs), the claim that the pair is minimal among all filling pairs is not fully supported.
- [§3] §3 (Description of the geometric procedure): the steps that lift the torus pair to S_g are presented via diagrams and local modifications, but the argument that coherence (i.e., geometric intersection equals |algebraic intersection|) is preserved after each modification is only sketched. A line-by-line verification that each local change preserves the equality would strengthen the central claim.
minor comments (2)
- [p. 2] The notation for the algebraic intersection number is introduced without an explicit formula or reference to the standard definition on page 2; adding the formula would improve readability.
- [Figure 2] Figure 2 (the main construction diagram) uses several unlabeled arcs; adding labels or a caption that identifies each arc with the corresponding curve segment would clarify the lifting steps.
Simulated Author's Rebuttal
We thank the referee for their careful reading and constructive comments. We address the two major comments point by point below.
read point-by-point responses
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Referee: [§4] §4 (Proof that the output pair is minimal): the manuscript shows that the constructed pair has a small intersection number obtained by the geometric lifting procedure, but does not recall or cite the known lower bound on the minimal intersection number of any filling pair on S_g. Without an explicit comparison showing that the output intersection equals this lower bound (rather than merely being smaller than some other pairs), the claim that the pair is minimal among all filling pairs is not fully supported.
Authors: We agree that the manuscript should explicitly recall and cite the known lower bound on the geometric intersection number of any filling pair on S_g. In the revised version we will add a reference to the established lower bound (from the literature on filling pairs) together with a direct verification that the intersection number produced by the construction equals this bound. revision: yes
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Referee: [§3] §3 (Description of the geometric procedure): the steps that lift the torus pair to S_g are presented via diagrams and local modifications, but the argument that coherence (i.e., geometric intersection equals |algebraic intersection|) is preserved after each modification is only sketched. A line-by-line verification that each local change preserves the equality would strengthen the central claim.
Authors: We accept that the preservation of coherence is only sketched in the current draft. The revised manuscript will expand the argument in §3 to include an explicit, step-by-step check that each local modification in the lifting procedure preserves the equality between geometric and absolute algebraic intersection numbers. revision: yes
Circularity Check
No circularity; construction is self-contained geometric lift from torus base case.
full rationale
The paper describes an explicit geometric procedure that starts from a known coherent filling pair on the torus (intersection number 1) and produces pairs on S_g (g≥3) asserted to be filling, coherent, and minimal. No equations or steps reduce the minimality claim to a fitted parameter, self-definition, or self-citation chain; the torus input is independent and the output properties are claimed to follow from the geometry of the lift. The central claim therefore does not collapse to its inputs by construction.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Standard facts about algebraic and geometric intersection numbers on surfaces
- domain assumption Existence of coherent filling pairs on the torus
discussion (0)
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