arxiv: 2605.05506 · v2 · submitted 2026-05-06 · 🧮 math.MG

Recognition: 2 theorem links

· Lean Theorem

Closed polylines with fixed self-intersection index

Dmitri Fomin

Authors on Pith no claims yet

Pith reviewed 2026-05-11 01:00 UTC · model grok-4.3

classification 🧮 math.MG

keywords closed polylineself-intersection indexpolygonal chaincombinatorial geometrycrossing numberexistence theoremparity conditionnon-existence

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The pith

Closed polylines exist in which every one of the n edges is crossed exactly k times, for every k and all sufficiently large n making nk even.

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

The authors determine exactly which pairs of positive integers n and k admit a closed polyline with n edges in the plane such that each edge crosses the polyline at exactly k interior points. For k equal to 1 or 2 the answer has been known for decades, but the paper supplies the full list of solutions when k is 3 or 4 along with general theorems ruling out some configurations. The final result is that the obvious parity requirement nk even is also sufficient once n exceeds a threshold that depends on k alone. Readers interested in combinatorial geometry would care because the work nearly settles the question of which uniform self-intersection patterns are realizable by straight-line closed chains.

Core claim

A closed polyline with n edges has the fixed self-intersection index k when each edge is crossed exactly k times by the other edges. The paper proves that such polylines exist for k=3 and k=4 precisely when n belongs to certain explicit infinite families satisfying the parity condition, and that for any k the polylines exist for all sufficiently large n with nk even.

What carries the argument

The fixed self-intersection index k, which counts the transverse crossings of each edge with the rest of the closed polyline.

If this is right

For k=3 the possible values of n are completely characterized.
For k=4 the possible values of n are completely characterized.
Some small n with nk even do not admit any such polyline.
For each k only finitely many n with nk even are impossible.
The constructions show that local crossing constraints can always be satisfied when n is large.

Where Pith is reading between the lines

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

Further work could determine the smallest possible n for each k to give the tightest bound on existence.
Analogous existence questions can be asked for open polylines or for self-intersecting chains with non-uniform crossing counts.

Load-bearing premise

The only intersections are transverse crossings so that the number of times each edge is crossed is a well-defined integer that does not depend on the particular drawing chosen.

What would settle it

A successful construction of a closed polyline with 100 edges each crossed exactly 3 times would confirm the large-n existence, while an exhaustive search that finds no such polyline for n=20 and k=3 would test the completeness of the solution for k=3.

read the original abstract

We investigate the existence of closed polylines (also known as closed polygonal chains or self-crossing polygons) that intersect each of their edges the same number of times. The most general question in this corner of combinatorial geometry asks for all pairs $(n, k)$ such that there exists a closed polyline with $n$ edges, each intersecting the same polyline exactly $k$ times. For $k = 1$ and $k = 2$, this is a very simple question answered several decades ago. In this article, we present a complete solution for $k = 3$ and $k = 4$, as well as the proof of some non-existence theorems. In conclusion, we show that, for an arbitrary positive integer $k$, a polyline of the required type exists for any sufficiently large integer $n$ such that $nk$ is even.

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.

Desk Editor's Note private letter to a colleague

Fomin classifies exactly which n work for k=3 and k=4, proves some non-existence results, and shows existence for all large n with nk even.

read the letter

The core advance is the complete classification for k=3 and k=4: which n admit a closed n-edge polyline where each edge crosses the rest exactly k times, together with non-existence proofs for the impossible small cases. The paper also gives an asymptotic existence theorem that removes all obstructions beyond the obvious parity condition once n is large enough. Previous literature only settled k=1 and k=2, so these explicit results and the general large-n construction are new. The constructions appear to be combinatorial, built from periodic or adjustable zig-zag patterns that keep crossings transverse and uniform per edge. The non-existence arguments rest on straightforward double-counting and local constraints rather than circular definitions. The parity requirement follows immediately from total crossings equaling nk/2 and is handled cleanly. Minor soft spots include the need to verify that the case analysis for small k really exhausts all possible configurations without overlooking a non-generic embedding, and that the large-n construction scales without introducing accidental tangencies. Both look manageable from the setup. This is targeted work in combinatorial geometry on self-crossing polygons. Readers who care about crossing numbers or realizability questions will find the concrete classifications useful. It deserves a serious referee because the claims are specific, the methods are elementary and checkable, and the gap it fills is well-defined.

Referee Report

0 major / 3 minor

Summary. The paper studies closed polylines with n edges in the Euclidean plane such that each edge is crossed exactly k times by the rest of the polyline. After recalling the known cases k=1 and k=2, it claims a complete classification of admissible (n,k) pairs for k=3 and k=4, proves several non-existence results, and establishes an asymptotic existence theorem: for every fixed positive integer k there exists N such that a polyline with the required property exists whenever n ≥ N and nk is even.

Significance. The explicit solutions for k=3 and k=4 together with the parity-based non-existence obstructions and the large-n existence result constitute a concrete advance in combinatorial geometry. The parity condition is obtained by double-counting and is therefore robust; the constructions appear to rest on direct geometric arguments rather than on quantities defined circularly in terms of the target property.

minor comments (3)

§2: the definition of the self-intersection index per edge is stated only for transverse crossings; a brief remark on how degenerate tangencies or vertex incidences are excluded (or shown not to arise) would prevent ambiguity in the non-existence arguments.
Theorem 4.1 (k=3 case): the enumeration of admissible n appears complete, but the proof that no other n work relies on an exhaustive case analysis whose length makes it hard to verify without an accompanying diagram or table summarizing the forbidden configurations.
§6 (asymptotic existence): the construction for large n is sketched via a periodic perturbation; adding a single figure illustrating the base pattern and the perturbation step would make the argument immediately checkable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of its contributions, and the recommendation for minor revision. We are pleased that the explicit classifications for k=3 and k=4, the parity obstructions, and the asymptotic existence result were viewed as a concrete advance.

Circularity Check

0 steps flagged

No circularity; results rest on explicit constructions and counting arguments

full rationale

The paper solves the existence question for closed polylines with uniform self-intersection index k by providing explicit constructions for k=3 and k=4, non-existence theorems, and an asymptotic existence proof for large n (with nk even). The parity condition is obtained directly from double-counting total crossings (nk/2 must be integer), which is an independent combinatorial necessity rather than a fitted or self-defined quantity. No steps reduce by construction to the target claim, no parameters are fitted then renamed as predictions, and the cited prior solutions for k=1,2 are external results from decades ago with no author overlap or load-bearing self-citation chain. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified from the given text. The work consists of existence and non-existence proofs in combinatorial geometry.

pith-pipeline@v0.9.0 · 5440 in / 1063 out tokens · 45757 ms · 2026-05-11T01:00:12.407840+00:00 · methodology

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We investigate the existence of closed polylines ... each intersecting the same polyline exactly k times. ... type ⟨n×k⟩ is feasible if either k even and n≥2k+3, or k odd and n even with n≥8k+6.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 2.1. If both numbers n and k are odd, there is no polyline of type ⟨n×k⟩. Proof via intersection graph G having odd number of odd-degree vertices.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

2 extracted references

[1]

(2003),Zadacha 3.7 (in Russian)

Kovalji, A.K. (2003),Zadacha 3.7 (in Russian). Matematicheskoe Prosveschenie,7, p.190–193 2003

2003
[2]

(2019),Zamknutye samoperesekayuschiesya lomanye (in Russian)

Blinkov, A.D., Gribalko, A.V. (2019),Zamknutye samoperesekayuschiesya lomanye (in Russian). Kvant, 10, p.26–28, 2019 Boston, USA Email address:fomin@hotmail.com

2019