Closed polylines with fixed self-intersection index
Pith reviewed 2026-05-20 22:49 UTC · model grok-4.3
The pith
Closed polylines exist where each of the n edges crosses the chain exactly k times for k=3,4,6 and all large n with nk even.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
A closed polyline with n edges has self-intersection index k when each edge intersects the polyline in precisely k points. The paper supplies explicit constructions and non-existence proofs that settle existence for every admissible n when k equals 3, 4 or 6, and shows that for arbitrary k the same polylines exist for all sufficiently large n satisfying the parity condition that nk is even.
What carries the argument
The self-intersection index k of a closed polyline, together with the global parity condition that nk must be even, which governs both the constructions for small k and the asymptotic existence proof for any k.
If this is right
- For k=3,4,6 the admissible n are completely characterized by the parity condition nk even plus a finite list of exceptions.
- For every fixed k the required polylines appear once n exceeds some bound that depends only on k.
- The same parity condition nk even is necessary for any such polyline to exist, independent of k.
- Non-existence holds for certain small n when k=3,4 or 6 even though the parity condition is satisfied.
Where Pith is reading between the lines
- The constructions may adapt to give minimal-length examples or examples with additional constraints such as convexity of the convex hull.
- One could ask whether the same parity obstruction persists for smooth closed curves or for polylines in higher-dimensional space.
- The general-position assumption leaves open whether the existence results survive when triple points or tangencies are deliberately allowed.
Load-bearing premise
All crossings between distinct edges are transverse double points with no three edges meeting at one location and no tangencies.
What would settle it
An explicit closed polyline with five edges each crossing the chain three times, or a rigorous proof that none exists, would decide whether the claimed complete solution for k=3 holds.
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 $k = 3, 4, 6$, 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.
Referee Report
Summary. The paper investigates the existence of closed n-edge polylines in the plane such that each edge intersects the remainder of the polyline exactly k times. It claims a complete solution for the cases k=3,4,6 together with several non-existence theorems for small (n,k) pairs, and proves that for any fixed positive integer k such a polyline exists whenever n is sufficiently large and nk is even.
Significance. If the constructions and non-existence arguments are made rigorous, the work would resolve the uniform-crossing problem for three additional small values of k and establish an asymptotic existence result, extending the classical solutions known for k=1 and k=2. The explicit geometric constructions and combinatorial case analysis constitute the main technical contribution.
major comments (3)
- [§3] §3 (Constructions for k=3): the explicit drawings are asserted to realize exactly three crossings per edge under a general-position hypothesis, yet no perturbation lemma or separate argument is supplied showing that the given configurations can be realized with only transverse double points and without triple points or tangencies that would alter the per-edge crossing count.
- [§4] §4 (Non-existence theorems): the exhaustive case analysis for small n relies on the assumption that all intersections are transverse double points; without a prior result excluding or handling degeneracies, the enumeration does not fully rule out other configurations that might satisfy the uniform-k condition.
- [§5] §5 (Large-n existence): the inductive or recursive construction for large n inherits the same general-position requirement; a uniform perturbation argument applicable to all sufficiently large n would be needed to guarantee that the crossing number remains exactly k after closure.
minor comments (2)
- The notation distinguishing the polyline from its underlying point set could be introduced earlier and used consistently.
- Figure captions for the k=4 and k=6 constructions should explicitly state the number of crossings verified per edge.
Simulated Author's Rebuttal
We thank the referee for the careful reading and constructive comments on our manuscript. We address each of the three major comments below and will incorporate the suggested clarifications and additional arguments into the revised version to strengthen the rigor of the general-position assumptions.
read point-by-point responses
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Referee: [§3] §3 (Constructions for k=3): the explicit drawings are asserted to realize exactly three crossings per edge under a general-position hypothesis, yet no perturbation lemma or separate argument is supplied showing that the given configurations can be realized with only transverse double points and without triple points or tangencies that would alter the per-edge crossing count.
Authors: We agree that an explicit perturbation argument is needed to make the constructions fully rigorous. In the revised manuscript we will add a short lemma (placed at the beginning of §3) showing that any closed polyline whose edges meet the remainder in the prescribed number of points can be perturbed by an arbitrarily small amount so that all intersections become transverse double points while preserving the exact per-edge crossing count. The lemma relies on standard transversality results in the plane and applies uniformly to the finite number of explicit configurations presented for k=3. revision: yes
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Referee: [§4] §4 (Non-existence theorems): the exhaustive case analysis for small n relies on the assumption that all intersections are transverse double points; without a prior result excluding or handling degeneracies, the enumeration does not fully rule out other configurations that might satisfy the uniform-k condition.
Authors: The non-existence proofs in §4 proceed by exhaustive combinatorial enumeration under the standing general-position hypothesis stated at the outset of the paper. To address the referee’s concern we will insert a preliminary paragraph in §4 that invokes the same perturbation lemma introduced in §3, thereby justifying that any hypothetical degenerate configuration can be perturbed to a transverse one without decreasing the crossing number below k; consequently the enumeration still rules out the existence of a uniform-k polyline. If a degenerate example were to exist, its perturbation would yield a transverse counter-example, which the case analysis already excludes. revision: yes
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Referee: [§5] §5 (Large-n existence): the inductive or recursive construction for large n inherits the same general-position requirement; a uniform perturbation argument applicable to all sufficiently large n would be needed to guarantee that the crossing number remains exactly k after closure.
Authors: We accept that the inductive construction in §5 requires an explicit uniform perturbation step. In the revision we will augment the proof of the large-n existence theorem with a single paragraph that applies the perturbation lemma (now stated once in §3) to the output of the inductive step. Because the lemma is independent of n and k (provided nk is even), it supplies the required uniform guarantee that, for all sufficiently large n, the final closed polyline can be realized with only transverse double points while keeping exactly k crossings per edge. revision: yes
Circularity Check
No circularity; explicit constructions and combinatorial arguments are self-contained
full rationale
The paper's claims rest on explicit geometric constructions for k=3,4,6, exhaustive case analysis for non-existence, and an asymptotic existence argument for large n with nk even. No equations, fitted parameters, or self-referential definitions appear; the general-position assumption is a standard modeling choice for transverse crossings but does not reduce any derivation to its own inputs by construction. All steps are independent of self-citation chains or renaming of prior results, making the work self-contained against external combinatorial geometry benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Intersections of line segments in the Euclidean plane obey standard crossing rules and parity.
Reference graph
Works this paper leans on
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[1]
(2003),Zadacha 3.7 (in Russian)
Kovalji, A.K. (2003),Zadacha 3.7 (in Russian). Matematicheskoe Prosveschenie,7, p.190–193 2003
work page 2003
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[2]
Blinkov, A.D., Gribalko, A.V. (2019),Zamknutye samoperesekayuschiesya lomanye (in Russian). Kvant, 10, p.26–28, 2019 24 DMITRI V. FOMIN Addendum: Professor Knop’s Zoo All the polylines in this section were graciously communicated to the author by his friend and colleague Konstantin Knop. They are the fruit of a very successful collaboration between him an...
work page 2019
discussion (0)
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