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REVIEW 3 major objections 4 minor 8 references

Quantum topology without topology

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Quantum invariants are algebraic functors, hence no topology needed

desk verdict Useful, well-referenced lecture notes on quantum topology via category theory; honest about deferred proofs, but too drafty as is. read the letter →

arxiv 2506.18918 v1 pith:YOD2PYJZ submitted 2025-06-13 math.QA math.CTmath.RT

Daniel Tubbenhauer This is my paper
classification math.QAmath.CTmath.RT MSC 18M3017B3718M0557K16
keywords quantuminvariantsBrauercategoriesdiagrammaticalgebracategoricalReidemeistermovesmonoidaltanglesribbon
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

These lecture notes argue that quantum invariants of knots and links can be built entirely from categorical algebra, with no topological input required. The central proposal is to define a quantum invariant as a structure-preserving functor Q that sends one of several diagrammatically presented Brauer-type categories into a linear category. Each Brauer-type category is claimed to be equivalent to a topological category of tangles, cobordisms, or ribbons: Br is equivalent to 1Cob, qBr to 1Tan, oqBr to 1State, and orqBr to 1Ribbon. If this is right, then familiar topological relations such as the Reidemeister moves become algebraic relations among generators, and constructing an invariant reduces to defining a functor on generators and checking that the algebraic relations are preserved.

What carries the argument

The central machinery is the generator-relation presentation of Brauer-type categories. These are strict monoidal categories whose objects are words in a single generator •, whose morphisms are built from crossings, caps, and cups, and whose relations include the Reidemeister 2 and 3 moves and the braided-pivotal calculus. The paper uses the universal property of such presentations to construct functors to topological categories and to pass between algebra and topology.

What would settle it

Find two tangle diagrams that are isotopic in three-space but are not identified by the generator-relation presentation of the corresponding Brauer-type category, for instance by checking whether every oriented version of the Reidemeister 3 move is provable from the relations in (5F-4) or (5H-11); a counterexample would directly falsify the claimed equivalence.

Watch

Extended reading notes

Core claim

The paper's claim is that a quantum invariant is a structure-preserving functor Q : -Br → C, where -Br is any of the diagrammatically presented Brauer-type categories Br, qBr, oqBr, or orqBr, and C is a 'linear algebra like category' such as finite-dimensional vector spaces. The load-bearing equivalences are Br ≃⊗ 1Cob (Theorem 3D.10), qBr ≃β,⋆ 1Tan, oqBr ≃β,⋆ 1State (Theorem 5F.9), and orqBr ≃β,⋆ 1Ribbon (Theorem 5I.8). In each case, the diagrammatic category is generated by crossings and caps and cups, with relations that encode planar isotopy and the Reidemeister moves. Because the equivalence is monoidal, braided, and pivotal where applicable, any invariant defined purely from the algebraic presentation automatically becomes a topological invariant.

Load-bearing premise

The whole construction rests on the assumption that the algebraic relations in the Brauer-type categories are complete, meaning that every planar isotopy and every Reidemeister move of tangles follows from the categorical axioms and the imposed relations; if two isotopic tangles are not identified by those relations, the resulting invariants are not topological invariants.

Editorial extensions

If this is right

  • An invariant can be defined by specifying images of the generating crossings and caps and by verifying a finite list of algebraic relations, which is often easier than working with topological isotopy classes.
  • The equivalences show that the topological categories 1Cob, 1Tan, 1State, and 1Ribbon have algebraic models, so computations in these categories can be carried out diagrammatically.
  • The framework gives a uniform treatment of the Brauer, Temperley–Lieb, BMW, and oriented quantum Brauer categories as models for different kinds of tangles.
  • Because the target category C is arbitrary among 'linear algebra like' categories, one gets many quantum invariants by choosing different linear targets, including categories of modules or fusion categories.
  • The paper's categorical Rosetta stone suggests that notions such as traces, dimensions, and twists in pivotal and braided categories directly yield invariants of the corresponding topological objects.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The faithfulness direction of the equivalences is where the real topological content lives; the paper itself notes explicitly that these proofs are 'hard and painful' or only sketched, so the central claim is only as strong as the completeness of the Reidemeister calculus in the algebraic settings.
  • One testable extension is to search for new quantum invariants by varying the target category C beyond finite-dimensional vector spaces, for example to categories of chain complexes or to fusion categories, while keeping the same Brauer-type source.
  • The categorical presentation viewpoint suggests that proving a given move holds topologically is equivalent to proving it as an equation in a diagrammatic category, which could make some classical knot-theoretic questions amenable to rewriting techniques.
  • If the equivalences hold, then the distinction between framed and unframed invariants is captured purely by the presence or absence of the ribbon relation in the source category, which is a clean algebraic reformulation of a topological subtlety.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript is a set of lecture notes (v1.99) covering categorical algebra from categories through monoidal, pivotal, braided, ribbon, additive, abelian and fusion categories, culminating in chapters on quantum invariants via diagrammatic categories and on growth and big-data approaches. The central thesis, stated in Section 5L, is that the diagrammatic Brauer-type categories Br, qBr, oqBr and orqBr are equivalent to their topological counterparts 1Cob, 1Tan, 1State and 1Ribbon, so that a quantum invariant can be defined as a structure-preserving functor Q : -Br → C and computed from generators and relations alone. The notes present many generator-relation constructions, diagrammatic proofs, exercises, and references to the literature.

Significance. If the missing faithfulness arguments are supplied or explicitly delegated to the literature, the notes will be a useful and largely coherent exposition: they give explicit presentations of Sym, TL, Br, qBr, oqBr, orqBr and Web, prove many diagrammatic identities, include numerous worked examples and exercises, and are honest about the difficult parts of the Reidemeister calculus. The manuscript is a survey and teaching text rather than a new research theorem, and the mathematical statements it uses are standard results. A particular strength is the systematic use of generator-relation presentations and the explicit separation of algebraic symbols from topological placeholders. No fitted parameters or circular derivations are involved. The appendix with MAGMA computations is a further useful resource.

major comments (3)
  1. [§5F.9, §5G, §5L] Section 5F.9 states the central equivalence qBr ≃β,⋆ 1Tan and oqBr ≃β,⋆ 1State, but the proof sketch stops after reducing isotopy to △-equivalence and then asserts, without demonstration, that the relations in (5F-2) and (5F-4) suffice to realize all versions of the Reidemeister moves. This is exactly the faithfulness direction needed for Section 5L's definition of a quantum invariant as a functor Q : -Br → C. The unresolved marker 'To Do: correct form for Brauer' in Section 5G and Exercise 5M.4, which asks the reader to write down all implicit relations, underscore that the presentation is not complete. The author should either complete the argument, replace the sketch by a precise citation to a full proof, or explicitly state that the equivalence is being assumed as a standard theorem.
  2. [§3D.10(d)] Theorem 3D.10 establishes Br ≃⊗ 1Cob, the model for all later equivalences, but part (d) of the proof says faithfulness is 'hard and painful' and refers to Exercise 3H.2 rather than proving or citing the Reidemeister calculus for Brauer diagrams. Because the entire algebraic approach of the notes relies on this faithfulness, the draft needs at least a reference to a complete proof at that point.
  3. [§5I.8] Theorem 5I.8, the orqBr ≃β,⋆ 1Ribbon equivalence listed in the Section 5L table, is delegated to [CP94, Section 5.3] without a statement of which result there gives full faithfulness and preservation of the braided rigid structure. Since this is the last column of the table on which the definition of quantum invariants rests, a precise pointer would make the reliance explicit.
minor comments (4)
  1. [§6E.1] In Definition 6E.1, 'for all X, Y ∈ Cat' should read 'for all X, Y ∈ C'; as written the definition quantifies over the wrong category.
  2. [§5J] In the paragraph before Definition 5J.6, 'not quote the case' should be 'not quite the case'.
  3. [§5L] The 'Reidemeister 1' column of the table is ambiguous because Section 5H distinguishes classical, ribbon, and framed Reidemeister 1 moves; the table should indicate which version is meant.
  4. [§2C] Convention 2C.1 introduces 'XY = X ⊗ Y' before the monoidal product is formally defined; a forward pointer to Definition 2D.1 would help readers.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the notes' central equivalences are cited as standard theorems or left as explicitly hard proofs; the gaps are expository incompleteness, not self-referential reductions.

full rationale

The load-bearing content is the claimed equivalences Br ≃ 1Cob, qBr ≃ 1Tan, oqBr ≃ 1State, orqBr ≃ 1Ribbon, summarized in Section 5L as 'the point is that they are all equivalent to their topological incarnations while "free XYZ with properties ABC". Thus, we define: A quantum invariant Q is a structure preserving functor Q : −Br → C.' These equivalences are not derived from the definition of the Brauer-type categories; they are imported from known topology. The paper itself marks the hard direction as unproved: Theorem 3D.10(d) says 'The proof that R is faithful is hard and painful, because one needs to show that the topologically defined 1Cob has the Brauer relations Equation 3D-8 as generating relations'; Theorem 5F.9 gives only a sketch ending with 'The final thing to check is that oqBr has enough relations to obtain all versions of the Reidemeister moves as well as all possible planar isotopies. Again, this is non-trivial'; Theorem 5I.8 is delegated to '[CP94, Section 5.3]'; and Section 5G contains the unprocessed marker 'To Do: correct form for Brauer'. These are genuine gaps in the notes as a self-contained derivation, and they are flagged here as correctness/completeness risks. They are not circularity: the algebraic categories and the topological categories are defined independently, the functors in the easy direction are checked against relations, and faithfulness is not assumed into the definitions. The self-citations [Tub23] and [Tub21] are disclosure of the source of exposition for web categories and lecture playlists, not load-bearing support for the central equivalences. There is no fitted parameter renamed as a prediction, no uniqueness assertion imported from the authors' own prior work, and no ansatz smuggled in by self-citation. Accordingly, the paper receives a low score reflecting only minor non-load-bearing self-reference, not circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The notes are a review, so the ledger is light: no free parameters (nothing is fitted to data) and no invented entities. The main load-bearing inputs are the known equivalences between diagrammatic and topological categories, Mac Lane coherence, the Duality Principle and the Freyd-Mitchell theorem, all standard or sketched with references. The author's own [Tub23] feeds the web-category exposition, but the underlying results are external to this paper.

assumptions (4)
  • domain assumption The diagrammatic categories Br, qBr, oqBr and orqBr are equivalent to their topological counterparts 1Cob, 1Tan, 1State and 1Ribbon, including the faithfulness of the functors R, qR, oqR and orqR.
    This is the load-bearing bridge of the notes, stated in Theorems 3D.10, 3G.5, 5F.9 and 5I.8. Faithfulness is admitted to be 'hard and painful' (Theorem 3D.10(d)) and is only sketched or deferred to [CP94] and [Pol10].
  • standard math Mac Lane's coherence theorem: every formal diagram of associators and unitors commutes (Theorem 2G.2).
    Used to justify strictification (Theorem 2I.5) and the graphical calculi; the proof in the notes is a sketch via Stasheff polytopes with π1(Kn) = 1.
  • standard math The Duality Principle metatheorem (Theorem 1C.1): a property holds in all categories if and only if its dual does.
    Invoked throughout to omit dual statements; the notes' 'proof' is the remark 'This is an experimental fact', which is acceptable for a metatheorem but is informal.
  • standard math The Freyd-Mitchell embedding theorem (Theorem 6H.6): every abelian category embeds into a module category.
    Cited with a sketch and reference [Fre64]; it underlies Section 6's claim that linear algebra can be run inside abelian categories.

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Cite this review

Pith. "Pith review of Quantum topology without topology." pith.science (2026). https://pith.science/paper/YOD2PYJZ

@misc{pith2026250618918,
  author       = {Pith},
  title        = {Pith review of: Quantum topology without topology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YOD2PYJZ}},
  note         = {Machine review of arXiv:2506.18918}
}
read the original abstract

These lecture notes cover 13 sessions and are presented as an e-print, intended to evolve over time. Quantum invariants do more than distinguish topological objects; they build bridges between topology, algebra, number theory and quantum physics helping to transfer ideas, and stimulating mutual development. They also possess deep and intriguing connections to representation theory, particularly through representations of quantum groups. These lecture notes aim to illustrate how categorical algebra provides a framework for studying both algebra and topology. Specifically, they demonstrate how quantum invariants emerge naturally from a mostly categorical perspective.

Figures

Figures reproduced from arXiv: 2506.18918 by the authors.

Figure 1
Figure 1. The Rosetta stone: the top and middle texts are in ancient Egyptian using hiero￾glyphic and Demotic scripts, respectively, while the bottom is in ancient Greek. The decree has only minor differences among the three versions, so the Rosetta stone became key to deciphering Egyptian hieroglyphs. Picture from https://commons.wikimedia.org/wiki/File:Rosetta_Stone_BW.jpeg In the 1980s, we have witnessed the birth of a fas… view at source ↗
Figure 2
Figure 2. It does not matter what is modern and what is not modern. In the end we all love the same thing: Venn diagrams , The picture is a variation of https://xkcd.com/2769/. Convention 1A.2. There are some set theoretical issues with the definitions of some categories. For example, the objects of Set are all sets, but such a collection does not form a set. These issues are completely irrelevant for the purposes of these no… view at source ↗
Figure 3
Figure 3. A picture of Geiger–Marsden (Rutherford’s gold foil) landmark experiments from ∼1910. See, for example, https://en.wikipedia.org/wiki/Rutherford_scattering_ experiments. Picture from https://en.wikipedia.org/wiki/Rutherford_scattering_experiments [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (43 more)
Figure 4
Figure 4. Figure 4: The category Veck categorifies Z≥0 and a lot of its structures, like addition, multiplication, etc. A goal of these notes is to categorify the following notions [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: A proof without words: The sum 1 + 3 + 5 + 7 + 9 + ... + 2n − 1 is the square n 2 . Wikipedia (as in the link below) writes: “For a proof to be accepted by the mathematical community, it must logically show how the statement it aims to prove follows totally and inevita…
Figure 6
Figure 6. Figure 6: The first few Stasheff polytopes. The middle is, of course, again a pentagon. Picture from [Kap93] (i) the equality holds, i.e. we have a commuting diagram [PITH_FULL_IMAGE:figures/full_fig_p028_6.png]
Figure 7
Figure 7. Figure 7: Generators and relations are well-studied for groups and an efficient way to encode the information. Picture from the “What is...algebra?” playlist on [Tub21] [PITH_FULL_IMAGE:figures/full_fig_p032_7.png]
Figure 8
Figure 8. Figure 8: A cut through (“projection”) a soap foam. Note the trivalent vertices, and the various faces (4,5,6,7 gons). Picture from https://commons.wikimedia.org/wiki/File:2-dimensional_foam_(colors_inverted).jpg Theorem 3G.5. There exists a monoidal functor R: Web → TGr, • 7→ •…
Figure 9
Figure 9. Figure 9: In a nutshell, Morse theory is the idea that most point on a manifold have nicely behaved neighborhoods and only a few critical points (called Morse points) where the behavior changes drastically need to be studied. The pictures show a saddle point on a surface and its…
Figure 10
Figure 10. Figure 10: Left: Various embeddings of a string into three space. These are all the same object, say a rope, that only differ how they sit in space. Right: A real world knotting. Pictures from https://en.wikipedia.org/wiki/Knot_theory and https://www.dtubbenhauer.com/lecture-geo…
Figure 11
Figure 11. Figure 11: A twisted ribbon. Picture from https://upload.wikimedia.org/wikipedia/commons/0/09/Ribbon_knot_8_20.jpg Example 5I.7. The (generic) oriented ribbon quantum Brauer category orqBr is the braided pivotal category generated by one object • with relations R : = , = , inclu…
Figure 12
Figure 12. Figure 12: The essence of linear algebra. Picture from the “What is...linear algebra?” playlist on [Tub21]. 6B. A motivating example. As we have already seen, the “multiplication” ⊗k of Veck generalizes to the notion of monoidal categories. Let us now focus on the “addition” ⊕ o…
Figure 13
Figure 13. Figure 13: Left: homology of a torus = swim ring; right: homology of a solid torus = donut. Picture from the “What is...algebraic topology?” playlist from [Tub21]. Definition 6I.2. A sequence (Xi , fi) • ∈ C is called exact in i if Ker(fi) = Im(fi−1), and exact if its exact in i…
Figure 14
Figure 14. Figure 14: The simple/indecomposable objects play the role of elements in our story. A main goal in abelian categories is to finite the corresponding periodic table. Picture from https://en.wikipedia.org/wiki/Periodic_table#/media/File:Colour_18-col_PT_with_labels.png Definition…
Figure 15
Figure 15. Figure 15: Quaternion plaque: “Here as they walked by on the 16th of October 1843 William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion mul￾tiplication i 2 = j 2 = k 2 = ijk = −1 & cut it on a stone of this bridge” Picture from https://en.…
Figure 16
Figure 16. Figure 16: n = 5 and n = 6 in Example 6J.13 illustrated geometrically. Picture from https://en.wikipedia.org/wiki/Quaternion Note that any (“finite”) X ∈ C with C ∈ Cat⊕, by definition, decomposes additively into indecomposables. However, Example 6J.5 shows that it is too much t…
Figure 17
Figure 17. Figure 17: In standard Sudoku, the goal is to populate a 9-by-9 grid with digits such that each row, each column, and each of the nine 3-by-3 subgrids contains every digit from 1 to 9 exactly once. Listing all groups (up to isomorphism or permutation) is similar to listing all S…
Figure 18
Figure 18. Figure 18: A crosswalk buttons is an idempotent: pressing it once, twice or a million times gives the same result. Picture from https://en.wikipedia.org/wiki/Idempotence • We calculate that we have a commuting diagram Z Im(eX) ⊕ Im(idZ − eX) Z Im(eX) ⊕ Im(idZ − eX) ( eX idZ−eX )…
Figure 19
Figure 19. Figure 19: Rarity does not inherently equate to dullness; in fact, it can often attract signif￾icant interest. Picture from https://www.chiesi.com/img/aree/sbmsjelqmcmalattie-rare-chiesi-farmaceutici.jpg 7F. Even more diagrammatics. Let us revise the categories TL, see Example 3…
Figure 20
Figure 20. Figure 20: Burnsides’s p a q b theorem implies that noncyclic simple groups have order divis￾ible by at least three primes. In particular, since the is no simple group of order 2 · 3 · 5 = 30, the smallest possible order for such a group would be 2 2 · 3 · 5 = 60. (Do you know a…
Figure 21
Figure 21. Figure 21: Some NP complete problems; they all can be used to simulate one another and also all other NP problems. Under the (almost by everyone accepted) assumption that P is not NP, all of these are “way too difficult” to solve. Wild algebras can be thought of as NP complete p…
Figure 22
Figure 22. Figure 22: A randomly generated 10-by-10 matrix with nonnegative entries and a plot of its eigenvalues and the eigenvector for the rightmost eigenvalue. Picture from https://www.youtube.com/watch?v=SJ0A0ewTxg4 (i) The vertices are {1, ..., n}. (ii) There is an edge with weight m…
Figure 23
Figure 23. Figure 23: Note that Q(L) ̸= Q(L′ ) implies L ̸= L′ , but the converse is not true in general. Hence, if a quantum invariant (or any invariant really, but we have not defined what these are) sends two knots to the same value, then we cannot conclude anything. Thus, a good quantu…
Figure 24
Figure 24. Figure 24: The figure eight knot (this picture needs to be rotated 90 degrees when compared to the discussion in this section). We add an orientation and perform Equation 9E-6. The circles are then reorganized using Lemma 5I.4. Picture from https://en.wikipedia.org/wiki/Figure-e…
Figure 25
Figure 25. Figure 25: The stereographic projection identifies C = R 2 with the Riemann sphere S 2 ∼= PC1 minus the northpole. Picture from https://dtubbenhauer.com/lecture-geotop-2023.html Algebraically speaking the Möbius group is PGL2(C) =  A =  a b c d [PITH_FULL_IMAGE:figures/full_…
Figure 26
Figure 26. Figure 26: An example of a Möbius transformation. Left: One antiprojects the grid onto the sphere. Right: Now rotate the sphere and project back. The grid has now become distorted. Pictures from https://www.geogebra.org/m/GhaSJw3t, due to Juan Carlos Ponce Campuzano. Proof. It i…
Figure 27
Figure 27. Figure 27: for an early reference [PITH_FULL_IMAGE:figures/full_fig_p146_27.png]
Figure 28
Figure 28. Figure 28: Sierpinski’s triangle is Pascal’s triangle modulo 2, and one gets a fractal type pattern. The fully black rows correspond to a = 2k − 1, since e.g. 7 = [1, 1, 1]2 so that the first case in Equation 10G-5 always applies. Pictures from https://www.youtube.com/watch?v=vW…
Figure 29
Figure 29. Figure 29: Webs are planar trivalent graphs. The name comes from the fact that they tend to look like spider webs. Picture from https://www.youtube.com/watch?v=-idEadSMnec 11B. Symmetric GLn webs. Similarly to Example 3G.2 we define the following. Recall that we think of the gen…
Figure 30
Figure 30. Figure 30: The prime number theorem: the number of primes π(n) is approximately n/ ln n: the error goes to zero, and the variance is an order of magnitude smaller than the actual values. (There is also an approximation using the logarithmic integral li(n), but we will ignore thi…
Figure 31
Figure 31. Figure 31: The first few trees showing that (tn)n∈Z≥0 = (1, 1, 1, 2, 3, 6, ...).) Picture created using AI that we have already seen in Example 8E.4. The PF eigenvalue and vector are λpf = ϕ, vpf = (1, ϕ). Now we normalize vpf to v ′ pf = √ 1 1+ϕ2 , which has length one. Now we …
Figure 32
Figure 32. Figure 32: Think of our problem of finding bn as the problem of counting how many planets (the summands) fit into the sun (V n). To get a precise growth rate of bn we would need to make a count about the average planet, call it almost-Uranus, and how many fit into the sun, but t…
Figure 33
Figure 33. Figure 33: Always double-check claims; or, more importantly, be sure of what you should be double-checking. Screenshot taken 10.Feb.2025. The story above marks the beginning of the paper [TZ25a], which we will summarize in this section [PITH_FULL_IMAGE:figures/full_fig_p220_33.png]
Figure 34
Figure 34. Figure 34: List of the first few prime knots (not including mirror images). Picture from https://en.wikipedia.org/wiki/List_of_prime_knots. see [OEI23, A002863]. Starting with the trefoil, every knot is listed explicitly. This is also the list we used for the big data comparison…
Figure 35
Figure 35. Figure 35: Average time; copyable data [PITH_FULL_IMAGE:figures/full_fig_p239_35.png]
Figure 36
Figure 36. Figure 36: Percentages of unique values; copyable data. n A2 A B1 J K KT1 J+KT1 All 4 1.0 0.9 0.9 1.0 1.0 1.0 1.0 1.0 5 2.9 3.0 2.9 2.9 2.9 2.9 3.0 2.9 6 6.0 5.9 6.0 6.0 6.0 5.9 5.9 5.9 7 13.0 12.9 12.9 13.0 12.9 12.9 12.9 12.9 8 34.1 33.9 34.0 34.0 33.9 34.1 34.0 34.0 9 83.2 74…
Figure 37
Figure 37. Figure 37: Average comparisons until equal; copyable data. n A2 Alexander B1 Jones Khovanov KhovanovT1 3 2 1 1 1 1 1 4 2 3 1 1 1 2 5 2 3 2 2 1 2 6 2 5 2 3 2 3 7 2 9 5 4 3 5 8 4 13 17 9 5 8 9 4 23 40 13 7 12 10 6 37 76 21 11 20 11 9 59 202 34 18 34 12 11 109 517 61 31 58 13 22 16…
Figure 38
Figure 38. Figure 38: Maximal coefficient; copyable data [PITH_FULL_IMAGE:figures/full_fig_p240_38.png]
Figure 39
Figure 39. Figure 39: Maximal coefficient sum; copyable data. n A2 Alexander B1 Jones Khovanov Khovanovt1 3 7.0000 3.0000 9.0000 3.0000 4.0000 4.0000 4 6.0000 4.0000 8.0000 4.0000 5.0000 5.0000 5 7.0000 5.0000 8.5000 5.0000 6.0000 6.0000 6 7.0000 7.5714 12.7142 7.5714 8.5714 8.5714 7 8.285…
Figure 40
Figure 40. Figure 40: Average coefficient sum; copyable data. n A2 Alexander B1 Jones Khovanovt1 3 0.5384 1.0000 0.3913 0.7500 0.4444 4 0.5384 1.6666 0.3913 1.0000 0.5454 5 0.5384 2.3333 0.3913 1.1666 0.6153 6 0.5384 3.0000 0.6585 1.8571 0.9333 7 0.5384 5.0000 1.1276 2.6250 1.2941 8 0.7600…
Figure 41
Figure 41. Figure 41: Maximum average coefficient; copyable data [PITH_FULL_IMAGE:figures/full_fig_p241_41.png]
Figure 42
Figure 42. Figure 42: Maximal spread; copyable data. n A2 Alexander B1 Jones Khovanovt1 3 13.0000 3.0000 23.0000 4.0000 9.0000 4 15.0000 3.0000 26.0000 4.5000 10.0000 5 16.5000 3.5000 30.5000 5.2500 11.5000 6 18.7142 3.8571 35.0000 6.0000 13.0000 7 21.2857 4.2857 41.0000 7.0000 15.0000 8 2…
Figure 43
Figure 43. Figure 43: Average spread; copyable data. n A2 Alexander B1 Jones Khovanovt1 3 1.1278 1.0000 1.0731 1.2106 1.1168 4 1.1837 2.6180 1.1080 1.2106 1.2406 5 1.1837 2.6180 1.1388 1.2837 1.2451 6 1.2196 2.6180 1.1990 1.6355 1.3042 7 1.2547 3.3165 1.2197 1.6355 1.6993 8 1.4348 4.3902 1…
Figure 44
Figure 44. Figure 44: Maximal absolute root; copyable data [PITH_FULL_IMAGE:figures/full_fig_p242_44.png]
Figure 45
Figure 45. Figure 45: Percentage of pure roots; copyable data. n A2 Alexander B1 Jones Khovanovt1 3 33.3333 100.0000 36.3636 0.0000 0.0000 4 57.1428 50.0000 56.0000 57.1428 11.1111 5 58.0645 80.0000 61.0169 23.5294 14.2857 6 54.8387 50.0000 65.5462 22.8571 16.6666 7 49.2957 65.2173 63.9285…
Figure 46
Figure 46. Figure 46: Percentage of roots in [0.9, 1.1]; copyable data. 13J. Exercises. Exercise 13J.1. Construct the A2 invariant using Section 11H. 3 Exercise 13J.2. Play with the data on [TZ25b]. 3 Exercise 13J.3. Above, we wrote “Examples of quantum invariants that satisfy a skein rela…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.