todo.md

To do list

Rough Draft

• Add theorems from BV to Gaußian chapter
• Add code (with comments) for the mma™ code.
• Fix toRVT code output for knots (should be long, not bent)
• Add output code to analysis section
• Add "further computations"
  • E.g. how the invariant interacts with satellites

Feedback from Dror

2023-06-01

Abstract

• Missing an executive summary; add to abstract or add an "executive summary" section.
• Provide the reader with a one-paragraph summary of the results of my thesis
  • Create invariant
  • to our surprise, it is distinct from the MVA
  • the structure is richer than Gaußians
  • &c.

Chapter: Introduction (Tensor products and Monoidal categories)

• symmetric monoidal / strict monoidal categories (mention terminology like this).
• Or, drop the entire statement about tensors.
• Scaring people away with fancy words is worse than hiding irrelevant terminology that an expert can connect with if they wish (cf. "associator", "braidor").
• Linear orders on sets
  • the linear order on a set is used for functionals. Dror finds this either confusing or wrong, and I am not so sure anymore.
  • I need to think more about how to fix this.
• The // notation needs to be clarified
• Clarify "Cartesian category". What does this mean? Add a footnote.

Chapter: Perturbed Gaußians

• Clean up RVT definition
  • hard to understand
  • no pictures are provided!
• 2.3 is just a summary of how to define the invariant. It is missing all the details.
• do proofs for the values of the generating functions

Chapter: Constructing the trace

• the trace has more properties; write them out
• explicitly define A_A as A/[A,A]
• What is a "shape"? The proof of lemma 3.4 is awkward. Clean it up.
• Here $\sim$ is undefined (equivalent modulo the coinvariants).
• "all the relations are accounted for" -> make the proof more formal.

The jump from the end of Constructing the trace to Conclusions is abrupt. Explicit examples are missing.

• Write a chapter in-between these two: Examples
  • compute the invariant of, say, the Hopf link
    • add pictures
    • add more details
  • add pictures
  • go through the computation
  • There is no statement connecting computations and conclusions.

"Pictures are hard to make. In this subject, they are necessary."

• hand-drawn can be time-consuming to modify.
• make pictures for Reidemeister moves, for whirling relations, &c.
  • Reidemeister moves
  • Whirling relations
  • toRVT algorithm

2023-06-26

• Address the rest of Dror's comments
  • Earlier feedback
  • 2023-06-26 feedback
    • p.10 wording changes
    • p.12 distinguish between vector space [A,A] and ideal [A,A].
    • p.13 grammar
    • p.15 math error
    • p.15 add notational description
    • Define a meta-object better; start with meta-monoids.
    • Remove meta-group reference (this is a meta-Hopf algebra).
    • Give a non-trivial example of a meta-monoid (say, n×n matrices)
    • p.21 Don't subscript integers.
    • p.23 Replace "pure" and "impure" with "open" and "mixed" (say).
    • The definition of framed tangles must put a condition on the endpoints.
    • Fix broken references.
    • p.27 mention in the Thm that this is only for virtual knots
    • p.36 "morphism-valued" is misleading.
    • p.39 Generating functions are not Gaußian in b-variables.
• Print the thesis off to read
  • Read over the thesis for a total of one time
  • Read over the thesis for a total of two times
  • Read over the thesis for a total of three times
  • Read over the thesis for a total of four times
  • Read over the thesis for a total of five times

Feedback from Joel

• 2.1 I found the notation T^D_C confusing. Does it denote a specific linear map from V_D to V_C or just an arbitrary one?

• 2.2 I didn't understand the definition of "meta-object", possibly because of missing words. Subcategory of what category? Is a meta-object a functor? Similarly, I don't understand equation (2.8). Similarly, I didn't understand the discussion afterwards about homsets and about the "map \sqcup". In the definition of meta-group (def 2.7), if X -> Y is a map of finite sets, do we have a morphism G_X -> G_Y ? For example, if X and Y both contain one element, do we have an isomorphism G_X \cong C_Y? What is the relation between "meta-group" and "meta-object"? And what is the relation between "meta-group in C" and "group object in C"? (Some of these same questions apply to "meta-algebra")

• 2.4

  • def 2.25: What does upward pointing arcs mean? I thought that the tangles are in a disc.
  • Theorem 2.28: is this a ... algebra in the category Set? Are these tangles considered up to isotopy?

• 2.5 What do you mean "The ribbon structure of U requires ..." ? maybe you should verify that these R, C satisfy the desired equations.

• 3.1

  • In Remark 3.2, I didn't understand what "extending the definition" means.
  • Why do you say that the categories \mathcal H and \mathcal U are isomorphic? If I understand correctly, if for example J, K are one element sets, then Hom_{\mathcal U}(J,K) = Hom(U, U) whereas Hom_{\mathcal H}(J,K) = Hom(Q[z], Q[z]) ? But U is a polynomial ring in 4 variables. I suppose that there is a vector space isomorphism U \cong Q[z] since both are countable dimensional, but there is no preferred one.
  • In Lemma 3.3, what is |_{z_K -> \del \zeta_K} ?

Feedback from Vassili

• The acknowledgements should fit into one page.
• Code annotations: Maybe a box around the annotation would have helped me? Or a very light background?

Feedback from Jesse

• The abstract should fit in one page.

Printout

• implement abstract comments
• implement acknowledgements comments
• add paper-vs-digital flag to tweak margins, page numbers, and colophon simultaneously
• § Executive summary
  • Understanding knotted objects
  • Make introductory picture more informative
  • "satellite", "ribbon", and "slice" should be introduced more consistently.
  • "properties" -> "operations"
  • add citation to Ribbon Slice Conjecture
  • Use wording from § 2.6 for building an invariant.
  • Add reference from § Quantum Invariants to relevant section.
  • Incorporate Quantum invariants tweaks
  • § Images of the Invariant
    • how about § Improving Computation Times
    • Add ref to relevant section.
    • Implement tweaks.
  • § Extending The Invariant...
    • Simplify title
    • add refs to appropriate sections
    • implement tweaks
    • modify open tangle image to be non-pure.
    • § Further Study -> make subsection instead of subsub? (match chapter outline)
      • Sentence is mentioned which is never elaborated on. Talk more about it!
• § Tensor Products and Meta-Objects
  • § Tensor Product Notation
    • Clean up language
    • Add reference to Penrose's book on the same notation.
    • change indices
    • clarify additional notation for tensor product
    • improve notation for indexed tensor product
    • Improve example to be easier to understand (use cross product ×).
    • Rearrange paragraphs to introduce notation in correct order.
  • § Meta-objects
    • Are subsubs necessary?
    • tweak language
    • add footnote
    • mention removal of "[X]" from notation
    • set bijections should be Greek
    • add "strict" and "symmetric" to monoidal categories
    • add actual matrices to Example: square matrices
    • add parentheses in intro sentence to meta-object definition
    • a meta-object is four things (or however many are actually listed).
    • refer to the appropriate figure for composition (×2).
    • add pronunciation of \then operator
    • Remove Remark 2.11?
  • § Algebraic Definitions
    • Add language tweaks
    • add more language to fig 2.1 caption
    • deal with undefined tensor notation in remark 2.13
    • add tensor product to remark 2.13
    • fix symbols used in fig 2.2
    • rem 2.14 needs associativity
    • fix rem 2.16
    • tweak text after rem 2.16
    • remove footnote "While notation explicitly..."
    • add meta- to meta-Hopf algebra definition
    • Remove "so-called"
    • tweak quasi-triangular meta-Hopf monoid definition.
    • tweak Drinfeld element.
    • Move/remove isolated proof.
    • put "distinguished grouplike element" into footnote
  • § Upright Tangles
    • Opening paragraph may not be clearest. Tweak.
    • Clean up Definition of framed tangle. Mention framing.
    • Figure Reidemeister 1 needs the word "framed"
    • add missing figure reference
    • tweak remark 2.30
  • § The meta-algebra structure of...
    • "We now formally connect…"
    • Add figure references to thm "tangles form a ribbon meta-Hopf algebra"
    • add figure representing the antipode
    • move remark 2.33 about virtual tangles after proof of thm.
    • fix missing reference in comultiplication footnote.
    • shrink multiplication figure to look less stretched.
    • associativity proof is brutal. Fix those sentences.
    • itemize the proofs various axioms
    • clean up move figure references in from theorem statement to proof.
    • spinner and anti-spinner figures should be subfigures
    • add "visualized in fig..." for quasi-triangular axioms
    • add "to show \tangle is ribbon" to ribbon proof.
  • § The ybax meta-algebra
    • Pick a consistent name, or at least provide multiple names consistently.
    • define B = e^-b
    • add "b-adic completion" to ybax / \CU definition
    • add second definition to ribbon element
    • specify that $f \in \polyring{\K}{b}$ OSTTE
    • add quotes around $a$.
    • add yields before eq 2.45
    • mention linearity of eq 2.45
    • Write Weyl commutation relation as a lemma w/ proof
      • find citation for lemma
      • move e^b coefficient in front
    • package ya_commute and xa_commute into lemma.
    • Exponential commutation relations doesn't jive with the b-adic topology. Resolve this issue.
    • eq. 2.50 and 2.52 are over wide.
    • Reword beginning of § 2.6.
    • morphisms between meta-objects are Capital Greek, e.g. Φ
    • add one more arrow to fig 2.21 (or add the algebraic objects to the figure caption).
    • Insert Z(K_(3,1)) in unreduced form as motivation for § 3: Perturbed Gaußians.
• § 3: Perturbed Gaußians
  • Introduction is redundantly worded. Clean it up.
  • Embed Remark 3.2 into Def 3.1
  • Further clarify the linear isomorphisms between H, U, and C.
  • Clarify why these three categories need to be mentioned.
  • Be clear about the use of pullbacks.
  • define contraction of generating functions explicitly (especially the composition of the ordering map and the generating function map / their inverses).
  • add missing word 'ribbon's in § Expressing Hopf algebra operations as perturbed Gaußians.
  • Should generating functions be distinguished from the original functions with bold somehow?
  • add references above equations in proofs of Thm 3.6 (meta-Hopf structure of U)
  • § Notational conventions
    • Remove first paragraph, and clean up next paragraph.
    • Replace out-of-nowhere example with continuation of the example in fig 2.21.
    • Remove second example. Replace with the general "computes the Alexander polynomial" sentence.
• § 4: Constructing the trace
  • § 4.1: extending an open tangle … : clean up name
  • clean up first paragraph.
  • clean up traced meta-algebra definition
  • remove coalgebraic compatibility
  • add commutation with operations on other strands
  • Lemma 4.3 : "as" -> "are"
    • tweak proof wording
  • lemma 4.4
    • add coinvariants caveat to lemma 4.4 (coinvairants as/are a trace map) (vector space, not ideal)
    • Remove undefined notation from lemma
    • Remove undefined notation from proof
    • Add to proof that trace commutes with other operations.
  • § a generating function for the coinvariants
    • the first paragraph should follow from the previous sections better
    • add parenthesis.
    • add Remark about formula missing ab term
      • (then add such a term maybe?)
  • § Computational examples
    • Add a picture and more computational details à la fig 2.21 reference in § Notational conventions, say of the Hopf Link.
• § 5: Conclusions
  • If mentioning the W-Lambert function, then provide an equation for it.
  • Eq. 5.2 is not an equation! Put its values inline, or make it a pair of equations.
  • § Further work is missing mention of restricting the universality of the trace to (perturbed) Gaußian frameworks.
• § A: Code
  • Add "Use case" subsub to Mathematica™ code.
  • be explicit about "former version" -> BN-vdV version.
  • p.60: We must also define...
  • p.65: Next are defined functions...
  • p.67: add "These match the quantities given in Thm 3.6."
  • § A.2: Implementation of the trace and Z^tr
    • Maybe "See ch 5" -> "see ch 4.2"
    • As described in ?? -> missing reference
  • § A.3
    • "consider the link in fig A.1"
    • "The crossings are absorbed in the order the knot's strand interact with them."
    • fig. A.2: front -> line
    • § Extending the algorithm
      • Remove first sentence; redundant
      • Explain Lemma A.1
    • § Use Case
      • Adjacent code blocks are possible with the new borders (×2)
      • footnote: "Upright Tangle" -> "upright tangle"
    • § Implementation
      • remove blank line before type Skeleton….
      • broken acronym (RVT)
      • remove blank line after PD.
      • "Only in this case will the function toRVT will then output…"
• § B: Table of values
  • It should be small
  • readable (i.e. fit on the page)
  • link to a table online (say on github)
• § Bibliography: no publisher for Kauffman's book?
• Ensure table of contents issues are resolved.
  • "Context" is a poor subtitle
  • Uniformize the algebra name (U(sl2+0), U, ybax, &c.)
  • Add sample computations to § Perturbed Gaußians
  • Add sample computations to § Constructing the Trace
  • Add (small) table of values
• Colophon: Add version number.

External Feedback

Comments and suggestions

• P.12, Section 2: It would be good to include references for “meta-structures” / “meta-objects”. (I can not see any in this section.)
• P.14, l.-2: I think that you should assume the category C to be monoidal. Otherwise, you should explain what {1} and M × M mean.
• P.16, Definition 2.4: It should be specified whether the isomorphisms MX ∼= MY (for any two finite sets X, Y ) are part of the structure. If this definition does not give all the relations in its fourth condition (which suggests the use of the words “In particular”), then a reference should be provided.
• P.16, Example 2.5: I am not sure that the property to be “symmetric” is required here for C.
• P.18, Definition 2.11: Is a “meta-algebra” a “meta-object” in the sense of Definition 2.8? If so, then Definition 2.11 should refer somewhere to this notion of “meta-object”.
• P.23, Definition 2.25: This definition of “open tangles” is too restrictive to support the operations that are described in Theorem 2.31 (compare with [BNvdVa]). For instance, the left-hand side of Figure 2.9 does not show a tangle in the sense of Definition 2.25: an arbitrary topological disk in the plane is not necessarily the “unit disk”.
• P.27: The operation ηi does not seem to be well-defined: where is this “new strand” added? This issue seems to be evoked on p.31 in the sentence “(For those worried that this equation depends on the location [...])”, without being resolved there.
• P.33, Remark 2.32: What does the adjective “adjacent” mean there? Besides, if one commits oneself “to only apply multiplication when doing so would result in a valid (classical) tangle” as stated here, then the collection {T up X }X does not seem to constitute (strictly speaking) a “meta-Hopf algebra” in the sense of Definition 2.18.
• P.34, Definition 2.33: It would be appropriate to give references for this ribbon Hopf algebra U. In particular, it would be nice to specify the exact relation between U and the (ϵ = 0)-reduction of the ribbon Hopf algebra D of [BNvdVa].
• P.35, (2.47): Given the way the degree completion U = ˆU(g) has been defined, one needs to assume that ξ belongs to bQ[[b]] for eξx to converge. A similar remark applies to (2.48) and (2.49).
• P.36, Lemma 2.37: This lemma follows from the same property for D proved in [BNvdVa], idem for Theorem 3.5 on p.41. Yet I agree that it is better to include direct proofs.
• P.37, l.-5: It is not clear why the invariant Z is well-defined by this procedure. One should either refer to the existing works in the literature on the “universal invariants”, or prove that the “ribbon meta-Hopf algebra” T up has the required universal property.
• P.40: The exposition could be improved here. Indeed, a category C is introduced at the top of the page, without defining its composition. The composition is only given in Lemma 3.2 at the bottom of the same page.
• P.42, l.4: What does “central” mean in Q[zK][[ζK]]?
• P.42, proof of Theorem 3.5: Is the proof of (3.14) addressed somewhere?
• P.46, Lemma 4.3: How is the closure performed? Does one assume that the two endpoints (of the strand to be closed) are “adjacent” in some sense?
• P.48, proof of Theorem 4.6: I do not understand the argument “inspection of the above comprehensive summary [...] that this set is indeed linearly independent”.
• P.50, Theorem 4.9: The notation µ in the formula for f(yi, bi, ai, xi) is in conflict with µ := (1 − e−α)¯zi.
• P.50, Remark 4.10: I do not understand where (4.21) comes from, and why the series S satisfies S = e−α−µS. This remark should be expanded.
• P.51 to p.54: Please explain why this part only considers “long links” and not “round links” by taking the trace along every component. For instance, why the value of tr1,2(Z(H)) for the Hopf link is not considered in §4.3?
• P.53, l.-1 & l.-7: Give a reference for the Thistlethwaite link table from which L5a1, L10a43, L7n2 are extracted, or, at least include pictures of those links.
• P.53, (5.1): It is surprising that the two partial traces give different values for the Whitehead link, where the two components play symetric roles.
• P.55, (A.1): How is the Whitehead link given here as an open tangle? (It seems that there is no symmetry, again, between the two components.)

Typos and other minor comments

• P.8, l.-8: “Each crossing assigned” → “To each crossing is assigned” (?)
• P.17, l.4: “Where the last column” → “where the last column”
• P.17, l.10: “such as a meta-colagebra” → “such as a meta-coalgebra”
• P.17, l.-3: φ[S] = φD C [S] goes the other way round: from AD⊔S to AC⊔S.
• P.18, l.4: “two spaces” → “two objects”
• P.18, l.6: “CD1⊔D2 → CC1⊔C2” → “AD1⊔D2 → AC1⊔C2”
• P.18, l.7: It is written “This is visualized in figure 2.1”, but this is not what I see in Figure 2.1.
• P.19, l.-10: “When C = (Vect, ⊗)” → “When C = (k − Vect, ⊗)”
• P.19, l.-8: “Then A∅ is a field” → “Then A∅ is the field k”
• P.20, l.4: “A meta-colagebra” → “A meta-coalgebra”
• P.22, l.-1: A reference seems to be missing in the sentence “The term we use is inspired by ??”.
• P.24, l.-4: “An upright tangle diagrams” → “An upright tangle diagram”
• P.25, l.8: I think that (to be consistent with the notation TX of the previous page) the notation T up X should refer to equivalence classes of tangles (and not tangle diagrams).
• P.26, l.-7: “figures 2.4 to 2.7” → “(figures 2.4 to 2.7)”
• P.35, (2.45): “ayr” → “ayr”
• P.37, l.-10: “is map” → “is a map”
• P.37, l.-9: “ϕX Y ” → “φX Y ”
• P.37, l.-5: “a U-valued tangle invariant” → “a U-valued tangle invariant Z”
• P.38, (2.57): The contribution of C4 seems to be missing in this computation.
• P.39, l.3: One verb to be removed in the proposition “(...) which develops uses (...)”.
• P.40, l.6: “Where zX” → “where zX”
• P.40, l.-3: “powerseries” → “power series”
• P.42, l.3: “z∗” → “ζ” (?)
• P.42, (3.12): It seems that the notation AJ has not been introduced yet.
• P.42, (3.20): It seems that all the −ξiηj should be changed to +ξiηj.
• P.43, (3.21): “+−” → “−” (twice)
• P.47, l.11: “adµ(u)” → “adµ(f)”
• P.47, l.-12: “The coinvariants of U” → “The space of coinvariants of U”
• P.47, l.-12: In this line and in the sequel, the variables a, x should be denoted by a, x.
• P.49, l.-3: The choice of the notations α, β, ξ, η is not good since they can be confused with the variables αi, βi, ξi, ηi “dual to” ai, bi, xi, yi.
• P.51, (4.23): “b1w2” → “b1a2” (?)
• P.51, (4.24): “b2w1” → “b2a1” (?)
• P.51, (4.25): Same comment as for (4.23) and (4.24).

Feedback from Vincent

• p.iv (end) : "fiancée" -> wife (Congratulations again <3 )
• p.7 (end) : "one longer one)." -> a longer one). (Sounds better?)
• p.8 (3rd par.) : "Each crossing assigned an element of AxA."
• p.9 (2nd par.) : "the value of Z on tables always take<s> the form of"
• p.11 (3rd par.) : "Firstly, determining what the relationship between ..." (something missing or extra?)
• p.11 (3rd par.) : Sentence starting with "Second is the challenge", kind of a long sentence?
• p.12 (3rd par.) : "u_iv_jw_k=v_ju_iw_k=w_ku_ju_i" (Shouldn't one of the last two u's be a v?)
• p.14 (4th par.) : Is reminding what concatenation means useful when you just defined it last paragraph?
• p.17 (2nd par.) : "Where the last column and row..." (Shouldn't w be lower-cased? As it's continuing the sentence.)
• p.19 (3rd par.) : "It is more common think of the unit as an element 1 \in V."
• p.20 (Def 2.14) : "meta-colagebra"
• p.22 (Def 2.23) : One parenthesis too many?
• p.22 (end) : You have a missing reference.
• p.23 (end) : I don't know about you, but I hate have a line change in an equation like "D x [-1,1]", let alone a page change.
• p.24 (start) : "Two open tangles are considered equivalent if there exists an isotopy of the embedding<s> ..."
• p.24 (2nd par.) : "... which one may thin of as open tangles with ..."
• p.28 (end) : The sentence starting with "Associativity (equation (2.17)) holds, ..." feels like it's missing something.
• p.34 (4th par.) : "This generalization of tangles deals exactly with the issue that multiplication need<s> not ..." (not 100% sure)
• p.37 (Def 2.38) : "A morphism \Phi between these meta-objects is map ..."
• p.39 (start) : "... which develops uses perturbed Gaussians to compute Z quickly." (One word too many?)
• p.40 (start) : "... we define categoris U, H, and C with objects finite sets and morphisms:" (I get what you mean, and it's not grammatically wrong, but it's confusing. Maybe something like "where objects are finite sets, and morphisms are:"?)
• p.46 (Lem 4.4) : "Then define A_{S,L} := A_S x (A_A)_L." (Since the next sentence starts with "Then" too, I'd drop it for this one. Also, it might prevent the equation from being cut off at the end of the line.)
• p.47 (1st par.) : Since an operation \phi^D_C manipulates on a disjoint set of tensor factors than tr^i, ..." (First, I think "operations \phi^D_C manipulate" sounds better, but that's subjective. Second, the sentence doesn't feel right. Maybe something like "\phi^D_C and tr^i manipulate disjoint sets of tensor factors,"?)
• p.47 (eq. 4.5) : This might just be me, but I usually put a period at the end of my equations if it ends a sentence. This happens before (like with equation 4.4) but it's the first one I noticed.
• p.47 (3rd par.) : "We start with a result which simplifies working with coinvariants:" (This is SUPER subjective, but I don't like this sentence. WAY too many w's. I'd replace "which" with "that".)
• p.47 (Thm 4.6) : "The coinvariants of U, U_U, has basis ..." (I know U_U is one thing, but grammatically, your subject is plural here. I think you need either to change "has" by have, or write something like "The space of coinvariants...")
• p.48 (1st par.) : "Observing when f is a monomial in equation (4.12),..." (Is "Observing" the right word here?)
• p.48 (2nd par.) : "Finally, all that remains to show is this set is linearly independent." (Sounds better? Something like that?)
• p.53 (start) : Sentence starting with "For some inputs to the trace," (Feels like a verb is missing?)
• p.54 (3rd par.) : "... the form in equation (4.16) does not <have/need> to conform to the same structure."