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arxiv: 2606.06143 · v1 · pith:SM3OP5YUnew · submitted 2026-06-04 · 🧮 math.GT

Minimal Filling K-Systems of Curves

Hong Chang , Xiao Chen , Wujie Shen This is my paper

Pith reviewed 2026-06-27 22:52 UTC · model grok-4.3

classification 🧮 math.GT
keywords filling k-systemsminimal curve systemssurface topologygenus g surfacescurve intersectionsoriented surfacestopological configurations
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The pith

The exact minimal number of curves in a filling k-system on an oriented surface of genus g is determined for all positive integers k and g.

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

This paper establishes the precise smallest size of a filling k-system on an oriented surface of genus g. The result applies uniformly to every pair of positive integers k and g. A sympathetic reader would care because the determination supplies both an explicit construction achieving the bound and a proof that no smaller collection works. The outcome resolves the minimal-size question completely using standard definitions from the literature.

Core claim

We determine the exact minimal number of curves in a filling k-system on an oriented surface of genus g for any positive integers k and g.

What carries the argument

A combinatorial or topological derivation that produces a closed-form expression for the minimal count of curves.

If this is right

  • For every pair of positive integers k and g there exists a concrete integer N(k,g) that is the minimal possible size.
  • There exist explicit collections of curves on the surface that achieve exactly N(k,g) curves while forming a filling k-system.
  • Every filling k-system on the surface must contain at least N(k,g) curves.

Where Pith is reading between the lines

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

  • The closed-form expression for N(k,g) could serve as a benchmark when enumerating curve systems by computer on surfaces of moderate genus.
  • The same counting technique might extend to related questions about minimal configurations with bounded intersection numbers on surfaces with boundary.

Load-bearing premise

The standard definitions of filling k-system and oriented surface of genus g from prior literature are sufficient to admit an exact closed-form minimal count that can be derived combinatorially or topologically without additional constraints or exceptions for large k or g.

What would settle it

An explicit construction of a filling k-system on some genus-g surface that uses strictly fewer curves than the derived minimum, or a proof that no collection of that size can fill the surface.

Figures

Figures reproduced from arXiv: 2606.06143 by Hong Chang, Wujie Shen, Xiao Chen.

Figure 1
Figure 1. Figure 1: The complement of the red curve and the blue curve is a disk; hence, they form a filling [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plumbing. Definition 2.12 (Associated surfaces). Let T be a thickened multicurve. For each boundary compo￾nent of T , we glue a disk to it and obtain a compact surface. We denote this new surface by ST , and call it the associated surface of the thickened multicurve T . Proposition 2.13. Suppose a multicurve L has a total of i intersection points and the thickened sys￾tem T of L has b boundary components. … view at source ↗
Figure 3
Figure 3. Figure 3: Examples of stairs when k is odd [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Examples for odd k. In the boundary of the thickened system, the outer parts are treated as vertices; two vertices are joined if the corresponding parts lie in the same boundary component. When both n and k are odd, the outer part consists of two components (coloured yellow and blue); when n is even and k is odd, the outer part consists of a single component (coloured yellow). 6 [PITH_FULL_IMAGE:figures/f… view at source ↗
Figure 5
Figure 5. Figure 5: An example of a stair when k is even [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Examples for even k. In the boundary of the thickened system, the outer part contains n + 1 components: a left component (coloured red), a right component (coloured blue), and the remaining n −1 components, where the parts above and below are paired in order. When k is odd, the two examples shown above are St(5, 1) and St(4, 3) as shown in [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: , yielding a new thickened multicurve. This operation is called a Reidemeister II move. Note that around the two intersection points there are six componentsthree above and three below. After performing this surgery, the three components above are connected into a single component, and similarly the three components below are connected into a single component [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The loco and carriage operation. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: k = 1, n ≡ 0 (mod 4) and k = 1, n ≡ 1 (mod 4). 10 [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: k = 1, n ≡ 2 (mod 4) and k = 1, n ≡ 3 (mod 4). 3.3.2 The case when k ≥ 2 When k ≥ 3 is odd, the construction consists of one Construction I (as defined for k = 1) followed below by k−1 2 copies of Construction II on the basic block St(n, 2) (defined below). When k ≥ 2 is even, we use k 2 copies of Construction II on the basic block St(n, 2) (in the case k ≡ 2 (mod 4) a more specific construction is requir… view at source ↗
Figure 11
Figure 11. Figure 11: k = 2, n = 9. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: k = 3, n = 9. When k is an odd integer with k ≥ 3, we assemble k−1 2 copies of Construction II sequentially below a Construction I. This provides a procedure that, starting from St(n,k), applies a series of Reidemeister I and Reidemeister II moves and yields a thickened k-system with exactly two boundary components. The method of attaching Construction II below the existing structure is the same in the re… view at source ↗
Figure 13
Figure 13. Figure 13: n ≡ 2 (mod 4). • When k ≡ 3 (mod 4), we assemble k−3 4 copies of Construction II sequentially below a Construction I [PITH_FULL_IMAGE:figures/full_fig_p013_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: k = 3,n = 14 ≡ 2 (mod 4). • k is even. When k ≡ 0 (mod 4), we assemble k 4 copies of Construction II. When k = 2, we take the bottom n−2 4 four-row blocks and perform the basic construction in each of them, and then apply Reidemeister I moves to the top two rows. In this case, the squares of St(n, 2) are connected into two components. One component connects the leftmost and rightmost boundary components v… view at source ↗
Figure 15
Figure 15. Figure 15: k = 2, n = 10 ≡ 0 (mod 4). When k ≡ 2 (mod 4), we assemble k−2 4 copies of Construction II at the bottom of the previous construction in St(n, 2). 3. n ≡ 3 (mod 4) When k = 2, consider the parallelogram-shaped grid St(n, 2). Divide the small squares into n−3 4 four-row blocks at the bottom, leaving the top two rows of the small squares. In each lower block, the leftmost and rightmost four-step stair segme… view at source ↗
Figure 16
Figure 16. Figure 16: k = 2, n = 7 ≡ 3 (mod 4) and k = 3, n = 7 ≡ 3 (mod 4) [PITH_FULL_IMAGE:figures/full_fig_p016_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: k = 2, n = 8 ≡ 0 (mod 4). 3.4 Construction when n = 3, 4, 5 1. n = 5. We distinguish two cases depending on whether k is odd or even. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: k = 3 ≡ 3 (mod 4), n = 5 and k = 5 ≡ 1 (mod 4), n = 5. • k is even. We perform two Reidemeister I moves and four Reidemeister II moves in St(5, 2) as follows to complete the construction when k = 2, and then concatenate k 2 copies of this construction. • n = 4. First, from bottom to top, we take every two rows as a unit and perform a Reidemeister II move in the middle part of each unit. In the lowest unit… view at source ↗
Figure 19
Figure 19. Figure 19: k = 2, n = 5 and k = 4, n = 5 [PITH_FULL_IMAGE:figures/full_fig_p018_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: k = 3 ≡ 3 (mod 4), n = 4 and k = 5 ≡ 1 (mod 4), n = 4. 2. n = 3 and k ̸= 1 is odd. In this case, each row contains two small squares. At the lower endpoint of the common edge of the two squares in each row except the top two rows, we perform a Reidemeister I move. For the top two rows, we carry out one Reidemeister II move. The resulting thickened multicurve then has three boundary components. In the fina… view at source ↗
Figure 21
Figure 21. Figure 21: k = 5, n = 3. We now reconnect the previously removed intersection P to the region between the top two rows where the Reidemeister II move was performed(while simultaneously performing the corresponding plumbing surgery on the thickened multicurve). We call this special operation. We prove that this connects the two boundary components of the thickened multicurve into a single component. Before the specia… view at source ↗
Figure 22
Figure 22. Figure 22: k = 3, n = 3. 3. n = 3 and k is even. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: k = 4, n = 3. 4. n = 3 and k = 1. In this case, after plumbing two annuli, we need to perform an additional plumbing between a third annulus and each of the previous two annuli to obtain the desired thickened multicurve. The plumbing can be carried out in only two ways, as shown in [PITH_FULL_IMAGE:figures/full_fig_p020_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: The first case of k = 1, n = 3. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: The second case of k = 1, n = 3. 4 Proof of theorem 1.2 In this section we provide a proof of Theorem 1.2. Note that in the previous two sections we deter￾mined only the maximal genus of a surface that admits a filling k-system with n curves for given n and k, but the proof did not give an explicit construction for every possible genus. We fill this gap here. Specifically, we need to show that for every i… view at source ↗
Figure 26
Figure 26. Figure 26: Bigon candidates. The following lemma describes all possible situations in which a bigon could arise: Lemma 4.3. If there is a bigon after the surgeries, it must occur at one of the bigon candidates of the original stair. Proof. If we perform a Reidemeister I move at a crossing point Xi (marked by a blue cross in the figure), the two adjacent bigons Bi and Bi+1 merge into a single bigon. Suppose that afte… view at source ↗
Figure 27
Figure 27. Figure 27: Bigon candidates and configurations A,B,C,D,E,F. Therefore, in the final configuration no Reidemeister move crosses the boundaries of the rect￾angles except possibly Reidemeister I moves applied at the intersection points Xi+1,...,Xj−1 (as in configuration F in [PITH_FULL_IMAGE:figures/full_fig_p023_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Omitting Reidemeister moves when k = 3 and n = 9. The Reidemeister moves that can be omitted are in the right loco operations and the carriage operations in the four-row blocks, which are labelled in green. It remains to show that a union of rectangles is also impossible. First assume that k is odd. Suppose the union is Sj s=i+1 Bs with crossing points Xi+1,...,Xj−1. Then, as shown in Lemma 4.4, no Reidem… view at source ↗
read the original abstract

In this paper, we determine the exact minimal number of curves in a filling $k$-system on an oriented surface of genus $g$ for any positive integers $k$ and $g$.

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

1 major / 0 minor

Summary. The manuscript claims to determine the exact minimal number of curves in a filling k-system on an oriented surface of genus g, for arbitrary positive integers k and g.

Significance. If substantiated, an exact closed-form count for minimal filling k-systems would constitute a concrete combinatorial result in surface topology, potentially useful for questions about curve complexes and filling properties. The abstract, however, supplies neither the formula nor any derivation, so the potential significance cannot be assessed.

major comments (1)
  1. Abstract: the central claim of an exact closed-form minimal count is stated without any supporting derivation, formula, or argument. This prevents verification of whether the result follows from the standard definitions of filling k-systems.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their feedback. We address the single major comment below.

read point-by-point responses

  1. Referee: [—] Abstract: the central claim of an exact closed-form minimal count is stated without any supporting derivation, formula, or argument. This prevents verification of whether the result follows from the standard definitions of filling k-systems.

    Authors: Abstracts are conventionally limited to a high-level statement of the main result. The manuscript body supplies the explicit closed-form expression for the minimal cardinality of a filling k-system (as a function of k and g), together with the complete derivation and proof that this bound is achieved and is minimal. These arguments rely only on the standard definitions of filling systems and k-systems and are developed in full detail after the introduction. The abstract therefore does not impede verification; the full text does so directly. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper asserts an exact closed-form minimal count for filling k-systems on genus-g surfaces derived from standard prior definitions of those objects. No equations, fitted parameters, self-citations as load-bearing premises, or ansatzes are exhibited in the abstract or claim description that would reduce the result to its inputs by construction. The derivation is presented as holding uniformly from combinatorial or topological reasoning on the given definitions, making the central claim self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are visible from the abstract alone.

pith-pipeline@v0.9.1-grok · 5537 in / 935 out tokens · 15930 ms · 2026-06-27T22:52:18.466743+00:00 · methodology

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Reference graph

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