# Verification report — OWR-3385-009 (Izmestiev: is the graph of a 4-polytope linked in the boundary but linklessly embeddable?)

Verification date: 2026-10-10.

**Verdict.** The note gives a **negative answer**: there is no convex 4-polytope whose graph is linked in the
boundary of the polytope and has another embedding that is not linked. Scope:
- **Proved in the note.** Theorem 1.1: a piecewise linear embedding of the graph of a convex 4-polytope P in a
  piecewise linear 3-sphere in which the boundary cycle of every 2-face has a panel is the boundary embedding
  followed by a piecewise linear homeomorphism. Corollary 1.2: for the graph of P the following are equivalent:
  no minor in the Petersen family; an embedding without two disjoint cycles of odd linking number; the boundary
  embedding is flat; any two disjoint cycles have disjoint panels in the boundary. Corollary 1.3: the null-space
  embedding of the Colin de Verdière matrix that Izmestiev attaches to a 4-polytope is the boundary embedding.
- **Relies on cited theorems that are not re-proved.** The theorem of Robertson, Seymour and Thomas (flat if and
  only if linkless modulo 2 if and only if no minor in the Petersen family; the hard direction is the main theorem
  of their paper of 1995), Böhme's lemma in the form of statement (2.8) of that paper, Alexander's theorem on
  polyhedral 2-spheres, and Theorem 2.4 of Izmestiev. The two results of Robertson, Seymour and Thomas are used
  in the piecewise linear category (Section 2 of the note, paragraph "Category").
- **Not new mathematics beyond the answer.** Theorem 1.1 is Böhme's lemma combined with the uniqueness theorem of
  Georgakopoulos and Kim for 2-complexes in the 3-sphere (Remark 4.1); the direct argument of Steps 3 to 5 is
  given for completeness. The technique is that of Robertson, Seymour and Thomas (1993, statement (2.2); survey,
  statement (4.3)). No novelty is claimed beyond the answer to the problem.
- **Open.** The general question of Lovász and Schrijver on null-space embeddings of linklessly embeddable graphs.
  The note settles it only for the matrices of Izmestiev.

The note is unrefereed.

## Statement checked
- **Primary source.** Oberwolfach Report No. 2/2009, "Discrete Differential Geometry", Oberwolfach Reports 6 (2009),
  no. 1, 75–144, doi:10.4171/OWR/2009/02 (workshop of 11–17 January 2009, organised by A. I. Bobenko, R. Kenyon,
  J. M. Sullivan and G. M. Ziegler). The open problems were collected by G. Rote (pp. 106–110). Problem 7, by
  I. Izmestiev, is on p. 108; its references are on pp. 108–109.
  - Read in the publisher's file and in the copy on the web page of G. Rote (by each of the three verification
    runs).
  - The problem calls an embedded graph in the 3-sphere linked if it contains two disjoint linked cycles, and asks
    whether there is a convex 4-polytope whose 1-skeleton is linked as a graph in the boundary but has a different
    embedding that is not linked.
  - Background given there: for a graph with Colin de Verdière number 4, a conjecture attributed to Lovász and
    Schrijver (1999) and Lovász (2001) says that the null space of a Colin de Verdière matrix yields a non-linked
    embedding; a polytope as in the problem would refute it, by Izmestiev's theorem.
- **The two papers named in the problem.** Lovász and Schrijver (1999), Remark 2, p. 1019: for a 4-connected
  linklessly embeddable graph and a matrix of corank 4 with the sign pattern of the graph and one negative
  eigenvalue they suggest that the null space may yield a linkless embedding, and say that they do not know
  whether this is true. Lovász (2001), Section 8, item 4, p. 235: for a linklessly embeddable graph and a Colin de
  Verdière matrix he asks whether the embedding in the 3-sphere obtained from the null space is linkless. So the
  sources state a hope and a question; the word "conjecture" is that of the problem list. Both passages were read
  (Lovász 2001 completely, at the writing of the note).
- **Corpus record.** ulamai/UnsolvedMath, record OWR-3385-009, dataset version 1.6.0, status open. Its statement is
  the problem in other words ("another embedding" for "a different embedding").

## Readings
| Reading | Answer | Where |
|---|---|---|
| "linked" pair = odd linking number (L3) | no such polytope | Corollary 1.2 |
| "linked" pair = non-zero linking number (L2) | no such polytope | Corollary 1.2 |
| "linked" pair = no embedded 2-sphere separates the two cycles (L1) | no such polytope | Corollary 1.2, Lemma 5.1 |
| "linked" pair = the two cycles do not bound disjoint discs (L0) | no such polytope | Corollary 1.2, Lemma 5.1 |
| different readings for the boundary embedding and for the other embedding | no such polytope, for each of the sixteen combinations | Corollary 1.2, last sentence |
| the other embedding only a topological embedding | same answer | Section 5.2, last paragraph |
| any notion of a linked pair implied by (L3) and implying (L0), for instance "one of the two curves is not null-homotopic in the complement of the other" | same answer | proof of Corollary 1.2 |
| "two linked cycles" read as "two cycles forming a Hopf link" | outside the scope; in this reading the answer would be a trivial yes (C(6,4)), and no reader of the source would take it | not in the note |
| the general null-space question of Lovász and Schrijver | not answered; settled only for the matrices of Izmestiev | Corollary 1.3, Section 1.4 |
| cell decompositions of the 3-sphere instead of polytopes | not treated | Section 1.4 |

## Results in the paper
- **Theorem 1.1.** Statement as in the verdict. Proof in five steps: (1) Böhme's lemma, applied to the boundary
  cycles of the 2-faces (two of them meet in nothing, a vertex or an edge: Lemma 3.1), gives panels that meet only
  along the graph; (2) the 2-skeleton X embeds; (3) for every facet F the union of the panels of its 2-faces is a
  PL 2-sphere, and by Alexander's theorem and Lemma 3.3 (X minus the boundary of F is connected) one of its two
  sides contains no point of the image of X; (4) the empty sides of different facets are disjoint (Lemma 3.2);
  (5) extension over the facets, and invariance of domain.
- **Corollary 1.2.** The four equivalent statements, and the negative answer in the readings (L0) to (L3).
- **Corollary 1.3.** Izmestiev's matrices: the null-space embedding is the boundary embedding followed by a
  homeomorphism onto the unit sphere; hence not linked when the graph is linklessly embeddable.
- **Lemma 5.1.** The implications between the readings.
- **Lemma 6.1.** If every edge that avoids a vertex v lies in a facet through v, the boundary embedding is flat;
  proved without the theorems of Robertson, Seymour and Thomas. Examples: simplex, pyramids, prism over the
  tetrahedron, product of two triangles.
- **Examples with certificates.** C(6,4) (Gale's evenness condition; the triangles 135 and 246 are the only ones
  that are not 2-faces; parity of Conway and Gordon and of Sachs); the prism over the octahedron (an explicit K_6
  model with fifteen witness edges; two disjoint 6-cycles with linking number ±1, read off from two levels;
  Figure 1); the truncated 4-simplex (the pair of 10-cycles expected to be linked by Eppstein has linking number
  ±1 by computation; an explicit K_6 model, so that every embedding of its graph is linked).

## Computations (consistency test; programs and outputs in reproducibility/)
No proof depends on a computation. The programs decide, for explicit polytopes with rational vertices, whether
some pair of disjoint cycles has non-zero linking number in the boundary and whether the graph has a minor in the
Petersen family. The test concerns linking numbers; it does not establish flatness. Negative answers of the SAT
solver are not certified.
- **All combinatorial types with at most nine vertices** (second verification run): 275,478 types
  (1, 4, 31, 1,294, 274,148 with 5, ..., 9 vertices); 9,080 not linked and without minor (1, 3, 19, 302, 8,755);
  266,398 linked, each with a verified K_6-minor (0, 1, 12, 992, 265,393). No exception: a pair with non-zero
  linking number exists exactly when the graph has a minor in the Petersen family.
- **Where the lists come from.** Nine vertices: the ancillary file of M. Firsching's paper (arXiv:1803.05205),
  rational coordinates for all 274,148 types; completeness is his theorem. Up to seven vertices: enumerated by run
  B from Gale diagrams (36 types). Eight vertices: 1,294 pairwise non-isomorphic polytopes found by run B, as many
  for each f-vector as in the table of Firsching's paper (classification of Altshuler and Steinberg); so all types
  were processed, given the published numbers.
- **Original programs.** All 30,266 nine-vertex types with at most 26 edges (8,755 not linked; 21,511 linked, each
  with a K_6-minor), a sample of 6,000 of the others (all linked), all 495 eight-vertex types with at most 22 edges
  (302 and 193), all types with at most seven vertices, the 347 classes of 4-dimensional 0/1-polytopes (160 and
  187), named and random polytopes; 40,785 polytopes were recomputed in a second Schlegel diagram. For the 30,266
  nine-vertex types the verdict of the original programs and of run B is the same polytope by polytope.
- **Comparison of linking numbers.** On 498,389 pairs of disjoint cycles in 680 polytopes the values of the
  original programs and of run B agree up to one sign for each polytope.
- **Further families (run B).** 194 named polytopes with up to 36 vertices, all types with at most seven facets,
  1,279 of the types with eight facets, all types with nine facets and at most ten vertices, 2,400 random
  polytopes with 10 to 14 vertices: 306,927 polytopes in all, counted with repetitions of types; each of the
  292,737 linked ones has a verified minor; for 980 polytopes without a linked pair the absence of a minor (which
  follows from the theorem of Sachs) was not decided. The 41,866 non-polytopal 3-spheres with nine vertices,
  which are outside the scope of Theorem 1.1: 41,861 linked with a K_6-minor, 5 not linked and without minor.
- **Run A.** Three samples of 1,500 polytopes with at most ten vertices, same pattern; a vertex as in Lemma 6.1
  in 198 of 207 polytopes without minor; no knotted cycle detected in 564 polytopes without minor; Simon
  invariants ±1; the kernel of Izmestiev's matrix confirmed numerically for seven polytopes.
- **Third run** (programs of its own, written from the statements of the note; `independent_run_2/`). About 3.1
  million point sets (random subsets of small grids and similar sets, random Gale diagrams, vertex deletions)
  were sorted into combinatorial types: all 1, 4, 31 and 1,294 types with 5, 6, 7 and 8 vertices were found again,
  as many for each f-vector as in Table 6 of the arXiv version of Firsching's paper, and 300 types with nine
  vertices were taken as a sample. For these 1,630 polytopes the linking numbers of all 1,086,276 pairs of
  disjoint cycles were computed (42,057 non-zero) and the minors decided exhaustively: the rows of Table 1 for 5
  to 8 vertices are reproduced (1/0, 3/1, 19/12, 302/992); in the sample with nine vertices 117 are not linked and
  183 linked; no exception; no linked polytope has a vertex as in Lemma 6.1 (298 of the 302 eight-vertex types
  that are not linked have one). The 347 classes of 0/1-polytopes: 160 not linked, 187 linked. The lists with
  eight vertices are complete given the published numbers of types.
- **Controls.** A linked straight-line embedding of the graph of the prism over the tetrahedron, which is not a
  Schlegel diagram, is reported as linked and without minor by the original programs and by run B: the test can
  detect the configuration asked for in the problem.
- **The exact program of the note** (`writing_stage/check_note.py`, 60 checks, all passed): the certificates of
  Section 6 and every number of Section 7, except those of the third run, against the recorded outputs.
- **Re-run.** `run_quick.sh` repeats, in one process, the programs that need no external data and finish within
  a few minutes, and compares with the recorded outputs. A complete run (2026-10-10, 32 minutes) reproduced the
  recorded outputs in all 41 comparisons, up to the recorded running times; its log is `RERUN_LOG.txt`. The
  short mode was run again from the extracted archive, at the writing and by the third run: 25 comparisons, none
  different. The programs of the third run are repeated by `independent_run_2/run_r2.sh` (five comparisons).
- **A source discrepancy, and what is established.** The number of classes of 4-dimensional 0/1-polytopes under
  the symmetries of the cube is 347 in three independent enumerations of the orbits (the original programs, run
  B, the third run: 402 orbits of subsets of the cube, 347 of them full-dimensional), in entry A105231 of the
  OEIS, which refers to Aichholzer, and as the sum of the numbers of classes with 5, ..., 16 vertices in Tables 1
  and 4 of Chen and Guo (arXiv:1101.0410), with which the third run agrees for each number of vertices (17, 40, 54,
  72, 56, 50, 27, 19, 6, 4, 1, 1). The arXiv version of Ziegler's "Lectures on 0/1-polytopes" (math/9909177v1)
  says 349. The printed version of the lectures and Aichholzer's paper were not seen. The note uses 347 and says
  this (Section 7).

## Independent verification runs
The results were first obtained with proofs and with the programs in `original/`. Three independent
verification runs, all AI-assisted, followed on 2026-10-10, each with its own programs. Run A examined the
statement, the cited theorems and every step of the proofs; run B repeated the computations independently,
re-ran and read the original programs, and searched the literature. Both examined the first write-up of the
results, before the note was written. The third run examined the final text of the note (in the package its
programs are in `independent_run_2/`: it is the second round of verification).

| Item | Run A | Run B | Third run (final text) |
|---|---|---|---|
| Statement against the source; readings of "linked" | CONFIRMED | CONFIRMED (statement) | CONFIRMED (both files read again; one sentence rephrased) |
| Cited theorems of Robertson, Seymour and Thomas; Böhme's lemma | CONFIRMED_WITH_FIXES (cite the lemma in the form (2.8) of the 1995 paper only; piecewise linear category) | form of the lemma checked on the scan, p. 195 | CONFIRMED_WITH_FIXES (pp. 185–195 and the references of the 1995 paper and the announcement read again; the exact form of (1.2) of the 1995 paper to be stated) |
| Facts on polytopes, Lemmas 3.1–3.3 | CONFIRMED | not its part | CONFIRMED (also on 1,630 face lattices) |
| Theorem 1.1 | CONFIRMED_WITH_FIXES (proof correct; category in the statement; reason why the union of the panels of a facet is a PL sphere) | not its part | CONFIRMED (every step in the category in which it is stated) |
| Corollary 1.2 | CONFIRMED | not its part | CONFIRMED (all sixteen combinations of readings) |
| Corollary 1.3 | CONFIRMED (Izmestiev's Theorem 2.4 read; kernel checked numerically) | Theorem 2.4 checked | CONFIRMED (one step written out) |
| Remark 4.1 (Georgakopoulos and Kim) | — | Theorem 1.3 read | CONFIRMED (the 2-skeleton satisfies the hypotheses in the sense of their Section 2; the embedding built from PL panels is locally flat in their sense) |
| Lemma 5.1 | proved by this run | not its part | CONFIRMED |
| Lemma 6.1 | proved by this run with a shorter argument for cycles through v | not its part | CONFIRMED, all parts, with the new part (v) |
| Examples of Section 6; Figure 1 | certificates for C(6,4) and the prism over the octahedron | linking number and K_6-minor of the truncated 4-simplex | CONFIRMED (every certificate recomputed with its own programs; Figure 1 recomputed from the printed coordinates) |
| Computations | consistency tests with its own programs: no discrepancy | CONFIRMED (all 275,478 types with at most nine vertices; 306,927 polytopes; no discrepancy) | CONFIRMED as far as repeated: all types with at most eight vertices, 300 types with nine vertices, the 0/1-polytopes |
| Original programs; the program of the note | not its part | CONFIRMED (re-run with identical outputs; code read; linking numbers equal on 498,389 pairs) | `check_note.py` read and run (60 checks); `run_quick.sh short` from the archive: 25 comparisons, none different |
| Literature, novelty, credits | references quoted from memory to be verified | CONFIRMED_WITH_FIXES (credits to Eppstein, Georgakopoulos and Kim, the survey, Saran, Balinski; Theorem 1.1 follows from the literature; unread sources to be listed) | CONFIRMED_WITH_FIXES (searches repeated, nothing found; two statements about sources and one credit to be made exact) |
| Bibliography | — | — | CONFIRMED (24 DOIs at Crossref; two entries added) |

**Corrections required by runs A and B, all made.** Run A: (1) Böhme's lemma cited as statement (2.8), p. 195, of
the 1995 paper only, with the remark that the versions for flat embeddings do not suffice for the first sentence
of Theorem 1.1; (2) "piecewise linear" in the statements, and in Step 3 the reason why the union of the panels of
a facet is a PL sphere; the sentence about tame embeddings removed; (3) the remark on cell decompositions to be
proved or omitted: it is omitted; (4) statement numbers quoted from memory verified or removed (the books are
cited without statement numbers); (5) certificates that can be checked directly added (Section 6). Run B: credits
and references (Eppstein's blog post of 13 December 2014 and "Treetopes and their graphs", Section 6.1;
Georgakopoulos and Kim; the survey of Robertson, Seymour and Thomas; Foisy; Saran; Balinski; Grünbaum; a source
for the number 347); the explicit statement how Theorem 1.1 follows from the literature (Section 1.4, Remark 4.1);
the draft of Lovász's book as read by run B (Section 16.3; Sections 1.2 and 7 of the note); the truncated
4-simplex (Section 6.4); no directory names of the working machine in the saved outputs.

**Changes made when the note was written, after runs A and B.** Lovász (2001), which those runs could not read,
was read completely, and the description of the null-space question in Sections 1.1 and 1.4 follows it (a
question there, a hope in the paper of 1999). Part (v) of the proof of Lemma 6.1 (panel of a cycle through the
apex) is a new construction: a triangulated half-neighbourhood of the path of the cycle in the sphere S is lifted
towards the apex away from the path, and completed by a cone. The K_6 model of Section 6.4, `check_note.py`,
`make_figures.py` and Figure 1 were new.

**The third run: what it examined, and the result.** It examined the final text, and the points just named first.
- Statement: Problem 7 read again in the publisher's file and in the copy of the collector; the quotation is
  exact; the question is answered as posed and as intended.
- Proofs: all derived again line by line. No wrong statement and no gap was found. Part (v) of Lemma 6.1 is
  correct (the radial map is projective, not affine, on each triangle, which does not affect the argument); the
  K_6 model of the truncated 4-simplex is correct; Remark 4.1 is correct.
- Sources read again and compared with what the note says: the announcement of 1993 completely; pp. 185–195 and
  the reference list of the 1995 paper; pp. 126–127 and 133 of the survey; Georgakopoulos and Kim, Introduction
  and Section 2 (arXiv); Eppstein's blog post with its comments and Section 6.1 of "Treetopes and their graphs"
  (arXiv); Lovász and Schrijver (1999), pp. 1017–1020; Lovász (2001), Sections 1, 3, 4 and 8; Izmestiev (arXiv),
  Sections 1 and 2; Section 16.3 of the draft of Lovász's book; the OEIS entry; Chen and Guo (arXiv), the tables;
  Ziegler's lectures (arXiv), the passage on the number of classes; Firsching (arXiv), Tables 1 and 6.
- Its programs: see "Computations". Figure 1: the picture is the parallel projection of two concentric octahedra
  (scales 1 and 2.4), which is a Schlegel diagram based at an octahedral facet; the projections of C and D cross
  exactly four times; the over/under information is that of one viewer for all four crossings; C is on top at
  three of them; the signed sums are −1 and −1.
- Corrections it required, all made: (1) abstract: the consistency test is described exactly (two disjoint cycles
  with non-zero linking number exactly when the graph has a minor in the Petersen family) instead of "the
  equivalence was confirmed"; (2) Section 1.1: the definition of the source is paraphrased, only the quotation of
  nine words is literal; (3) after Theorem 2.1: the exact form of statement (1.2) of the 1995 paper (semi-flat)
  instead of "a condition of the kind (L0)"; (4) paragraph "Category": the proof of (2.8) refers to
  Rourke–Sanderson for general position; statement (1.7) of the 1995 paper cited for uniqueness; (5) Section 1.4:
  what Eppstein's texts say about K_7 and about the Petersen family; (6) the last step of the proof of
  Corollary 1.3 written out for the reading (L1); (7) statement (1.2) of the survey cited for the fact behind
  Lemma 6.1; (8) the number 347: what is established, with two new references; (9) the Verification paragraph and
  Section 7 brought to the final state; (10) the lists of what was read and of the searches updated, arXiv
  versions named; (11) the programs and outputs of this run added to the package; (12) one more example of a
  notion of "linked" in the proof of Corollary 1.2.
- Not checked by the third run: the proofs of the cited theorems (it read the proof of Böhme's lemma in Section 2
  of the 1995 paper, as run A had done); the completeness of the published lists; the computations for nine
  vertices beyond its sample.

**Not included in the note.** Run A proved a generalisation of Theorem 1.1 to regular cell decompositions of the
3-sphere in which the intersection of two closed cells is a closed cell or empty (its Proposition R), with an
example showing that this intersection property cannot be dropped. This remark was left out of the note.

## Relation to the literature, novelty and scope
- **Searches (10 October 2026; repeated by the third run on the same day).** arXiv, zbMATH, Crossref, OpenAlex,
  Semantic Scholar, DataCite and Zenodo; the lists of works citing Izmestiev's paper (18 in OpenAlex, 25 in
  Semantic Scholar), Lovász and Schrijver 1999 (40), Lovász 2001 (31), Lovász and Schrijver 1998 (76),
  Georgakopoulos and Kim (4 and 6) and Eppstein's paper (4 and 3); eight web searches. None of the titles and
  abstracts seen addresses Problem 7 or the boundary embedding of polytope graphs, apart from the works named
  below. The paper of Izmestiev and Lam of 2025 on discrete Laplacians was searched in full text: it does not
  mention the problem.
- **Closest prior work.** D. Eppstein, blog post of 13 December 2014 and "Treetopes and their graphs" (Discrete
  Comput. Geom. 64 (2020), Section 6.1 of the arXiv version): knots and links formed by the graph of a 4-polytope
  in its boundary; the smallest linked example (C(6,4)); pyramids and stacked polytopes contain no links; every
  link occurs in some 4-polytope. The post notes that a neighbourly polytope with seven vertices contains a knot
  because K_7 does in every embedding, and the paper names the Petersen family as forbidden minors for the graphs
  of pyramids; beyond this, these texts do not relate links in the boundary to linkless embeddability.
  A. Georgakopoulos and J. Kim, Theorem 1.3 (uniqueness of locally flat embeddings of simply connected 2-complexes
  with 3-connected link graphs). Robertson, Seymour and Thomas (1993), statements (1.4) and (2.2), and statement
  (4.3) of their survey; statement (1.2) of the survey (a planar graph with one further vertex is flat) for
  Lemma 6.1. Foisy et al. (2024) and Foisy (2022): other uses of Böhme's lemma (abstracts read).
- **What was read.** Problem 7 in the publisher's file and in Rote's copy; Robertson, Seymour and Thomas (1993)
  completely; of their paper of 1995, in a scan on the web page of one of the authors, pp. 185–197 (run A; pp.
  185–195 again at the writing and by the third run, with the list of references), the remaining pages, which
  contain the proof of the main theorem, by nobody; of their survey pp. 125–130 and 133–134; Lovász and Schrijver
  (1999), pp. 1017–1020; Lovász (2001) completely, in the publisher's file; Izmestiev (arXiv, version 3), Sections
  1 and 2, without the appendix on the signature; Georgakopoulos and Kim (arXiv, version 1), the introduction with
  Theorem 1.3 and Section 2; Eppstein's blog post with its comments and Section 6.1 of "Treetopes and their graphs"
  (arXiv, version 1); the passages on the open question in Schrijver and Sevenster (2017) and Lovász and Schrijver
  (2017), arXiv versions; the abstract, the introduction and the tables of Firsching's paper (arXiv, version 2);
  the tables of Chen and Guo (arXiv, version 1) and the passage on the number of classes in Ziegler's lectures on
  0/1-polytopes (arXiv, version 1); Section 16.3 of a draft of Lovász's book "Graphs and Geometry" dated February
  2019, on the web page of its author (it treats flat embeddings and Böhme's lemma, and remarks that skeletons of
  3-polytopes are flat and that skeletons of 4-polytopes may or may not be linklessly embeddable; it does not
  consider the embedding in the boundary).
- **Not read.** The survey of Ramírez Alfonsín (the publisher gave no automated access); the originals of Böhme,
  Saran, Sachs, and Conway and Gordon; the printed version of Lovász's book; the journal versions of the papers
  read in their arXiv versions; Aichholzer's paper on 0/1-polytopes; Altshuler and Steinberg (1985);
  Alexander (1924), Balinski (1961), Lovász and Schrijver (1998) and the books of Ziegler, Grünbaum, Rourke and
  Sanderson, Hatcher and Rolfsen, which are cited for standard facts. The proofs of the cited theorems were not
  checked, except that run A and the third run read the proof of Böhme's lemma in Section 2 of the paper of 1995.
- **Caveats.** The paper of 1995 assumes that all embeddings and subspaces are tame and argues with piecewise
  linear general position (for which it refers to Rourke and Sanderson); the note uses its results for piecewise
  linear embeddings and piecewise linear panels and says so. For the lemma this is the form in which it is proved
  there; for the main theorem it is the reading of the note, as in the announcement of 1993, where all spatial
  embeddings are piecewise linear. The statement of Saran's thesis is taken from the reference list of Robertson, Seymour and Thomas
  (1993). A search that finds nothing is not a proof of novelty.
- **Scope.** The note answers Problem 7 in the negative for the readings (L0) to (L3). It does not answer the
  general question of Lovász and Schrijver. No priority is claimed beyond the answer to the problem.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/