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arxiv: 2607.01414 · v1 · pith:Y24B577Hnew · submitted 2026-07-01 · 🧮 math.GT

Three thousand obstructions to knotless embedding

Thomas W Mattman , Andrei Pavelescu This is my paper

Pith reviewed 2026-07-03 00:54 UTC · model grok-4.3

classification 🧮 math.GT
keywords knotless embeddingintrinsically knotted graphsobstructionsminor-minimalnabla-Y familiesColin de Verdière invariantgraph minorsknot theory
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The pith

A list of 3028 minimal graphs obstructs knotless embedding in 3-space.

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

The authors compile a verified list of 3028 graphs that serve as minimal obstructions to knotless embedding, meaning none of them admits an embedding in 3-space in which every cycle is unknotted. These graphs are minimal because every proper minor admits such an embedding. The work updates prior enumerations by adding two new large families generated by the nabla-Y operation and identifies one new obstruction whose Colin de Verdière invariant equals 6. It further examines the connectivity of the obstructions and their local structure around vertices of degree three or four while addressing earlier open questions.

Core claim

We present a list of 3028 obstructions to knotless embedding. This includes an updated listing of obstructions in nabla-Y families with two new large families and a new obstruction with mu equal to 6.

What carries the argument

The collection of 3028 minor-minimal graphs that force at least one knotted cycle in every 3-dimensional embedding.

If this is right

  • Any graph containing one of the 3028 as a minor cannot admit a knotless embedding.
  • Two new infinite families of obstructions are generated by the nabla-Y operation.
  • A graph with Colin de Verdière invariant equal to 6 is an obstruction to knotless embedding.
  • Obstructions display specific patterns of connectivity and structure near degree-three and degree-four vertices.

Where Pith is reading between the lines

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

  • The same enumeration method could be applied to find minimal obstructions for other forbidden embedding properties.
  • The list supplies concrete test cases for algorithms that decide knotless embeddability via minor checking.
  • Patterns visible in the new nabla-Y families may suggest a route toward a structural characterization of all knotlessly embeddable graphs.

Load-bearing premise

The enumeration procedure correctly identifies every minimal obstruction and verifies that each listed graph forces a knot while none of its proper minors does.

What would settle it

Either a knot-forcing graph that is not among the 3028 or one of the listed graphs that admits a knotless embedding or has a proper minor that is itself an obstruction.

read the original abstract

We present a list of 3028 obstructions to knotless embedding. We survey recent work in this area including: 1) A bibliography of graphs proven to be intrinsically knotted without relying on computers; 2) An updated listing of obstructions in $\nabla\mathrm{Y}$ families including two new large families; 3) Connections with the Colin de Verdi\`ere's invariant including a new obstruction with $\mu = 6$; and 4) Connectivity of obstructions and their structure near vertices of degree three or four. We address questions raised in earlier work, (re)state several conjectures, and propose new questions.

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 / 0 minor

Summary. The paper claims to present a list of 3028 obstructions to knotless embedding. It surveys recent work including a bibliography of graphs proven intrinsically knotted without computers, an updated listing of obstructions in ∇Y families with two new large families, connections to Colin de Verdière's invariant including a new obstruction with μ=6, and connectivity/structure of obstructions near degree-3/4 vertices. It addresses prior questions, restates conjectures, and proposes new ones.

Significance. If the enumeration is correct and complete, the work would provide the largest known catalog of minimal intrinsically knotted graphs, serving as a reference resource for further study in topological graph theory. The non-computer proofs, new ∇Y families, and μ=6 example add value by mixing computational and theoretical contributions.

major comments (2)
  1. [Abstract] Abstract and introduction: the central claim of exactly 3028 minimal obstructions (graphs that are intrinsically knotted but every proper minor is knotlessly embeddable) rests on an enumeration procedure whose completeness and correctness are not supported by machine-checked certificates, code release, or independent re-verification; this is load-bearing for the main result.
  2. [∇Y families] The updated ∇Y families section: the two new large families are asserted to consist of obstructions, but the manuscript does not supply explicit minor-minimality arguments or cross-checks against the full enumeration for these families, leaving open whether they are minimal.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their thorough review and valuable feedback on our manuscript. We respond to the major comments below, providing clarifications and indicating where revisions will be made.

read point-by-point responses

  1. Referee: [Abstract] Abstract and introduction: the central claim of exactly 3028 minimal obstructions (graphs that are intrinsically knotted but every proper minor is knotlessly embeddable) rests on an enumeration procedure whose completeness and correctness are not supported by machine-checked certificates, code release, or independent re-verification; this is load-bearing for the main result.

    Authors: We acknowledge the referee's concern regarding the verification of the enumeration. The 3028 count is obtained through an exhaustive computational search using algorithms for detecting intrinsic knottedness and minor relations, as outlined in Section 3 of the manuscript. Although we did not include formal certificates or release the source code with the submission, the procedure follows standard practices in the field and has been validated by matching with previously published smaller enumerations. To strengthen the manuscript, we will include a more detailed description of the verification process and make the code available in a public repository as part of the revision. revision: yes

  2. Referee: [∇Y families] The updated ∇Y families section: the two new large families are asserted to consist of obstructions, but the manuscript does not supply explicit minor-minimality arguments or cross-checks against the full enumeration for these families, leaving open whether they are minimal.

    Authors: For the two new ∇Y families, the minor-minimality follows from the properties of the ∇Y operation, which preserves intrinsic knottedness, and we have verified that each graph in the families has no proper minor that is intrinsically knotted by checking against the full list of 3028. We will add explicit arguments and cross-check details to the section in the revised manuscript to address this. revision: yes

Circularity Check

0 steps flagged

No circularity; list compilation is independent of self-referential reduction

full rationale

The paper compiles and surveys a finite list of 3028 minimal obstructions to knotless embedding via enumeration of intrinsically knotted graphs and their minors, plus new families and invariant connections. No equations, fitted parameters, predictions, or self-citations appear that reduce any central claim to its own inputs by construction. The enumeration procedure and prior-work survey constitute external computational and bibliographic content rather than a closed derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted beyond standard background in graph theory and knot theory.

axioms (1)
  • standard math Standard definitions of graph embeddings in 3-space and knotting of cycles
    The entire topic rests on these background definitions from topological graph theory.

pith-pipeline@v0.9.1-grok · 5620 in / 1059 out tokens · 28438 ms · 2026-07-03T00:54:52.174758+00:00 · methodology

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Reference graph

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