Zesting and the relative complexity of Reshetikhin-Turaev invariants

Calvin McPhail-Snyder; Colleen Delaney

REVIEW 2 major objections 5 minor 6 references

Zesting and the relative complexity of Reshetikhin-Turaev invariants

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read When two ribbon categories are related by zesting, their Reshetikhin-Turaev link invariants are polynomial-time equivalent.

desk verdict A solid, genuinely useful paper proving that zesting preserves the polynomial-time complexity of simply colored Reshetikhin–Turaev link invariants, provided the zesting data is fixed and number-field valued; the unstated complexity hypothesis needs to be made explicit. read the letter →

arxiv 2608.02795 v1 pith:QPMDYWAL submitted 2026-08-03 math.QA math-phmath.GTmath.MP

Colleen Delaney , Calvin McPhail-Snyder This is my paper · ORCID

classification math.QAmath-phmath.GTmath.MP MSC 57K1618M2057K12

keywords zestingReshetikhin-TuraevinvariantsribbonfusioncategoriescomputationalcomplexityTuringequivalenceshadowrackcocyclehomotopyquantumfieldtheorylink

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

This paper proves that the zesting construction, which modifies a ribbon fusion category by cohomological data to produce a new category, does not change how hard it is to compute Reshetikhin-Turaev invariants of links colored by simple objects. If two categories are related by zesting, their link invariants are polynomial-time Turing equivalent: computing one from the other costs only polynomial overhead. The reason is a new tangle invariant, the zest invariant, which factors the zested invariant into the original invariant times a correction depending only on the zesting data, and which can be evaluated quickly from a link diagram. The result matters because exact quantum link invariants range across a wide complexity landscape, and a structural relation that guarantees equal complexity lets researchers transfer hardness or easiness between topological quantum field theories and organize topological phases by computational class.

What carries the argument

The central object is the zest invariant $J_\zeta$, defined as a 2-functor from the 2-category of $A$-shadow-colored, oriented, framed tangles to the delooping of the pointed subcategory of invertible objects. A shadow coloring labels each region of a tangle diagram by an element of the universal grading group $A$, with prescribed rules at strands and crossings. The invariant is computed locally: crossings contribute scalars $\phi_i(a,b)=\nu(i,b,a)^{-1}t(a,b)\nu(i,a,b)$, cups and caps contribute factors built from the zesting data $(\lambda,\nu,t,\epsilon)$, and closed loops contribute $\epsilon(a)$. This local calculus is what lets the paper evaluate $J_\zeta(L)$ in polynomial time: convert the link diagram to a braid closure with a polynomial-time algorithm, read off the shadow coloring, multiply the crossing factors, and finish with the cup and cap corrections. The factorization identity relating $J_\zeta$ to the Reshetikhin-Turaev functor is the load-bearing mechanism that turns a local invariant into a statement about complexity.

What would settle it

One concrete observation that would settle the claim: exhibit two ribbon fusion categories over a number field related by zesting and a family of simply colored link diagrams whose sizes grow polynomially but for which the local formula for $J_\zeta$ requires super-polynomially many arithmetic operations, or for which $F_{\mathcal{C}^\zeta}$ is provably not polynomial-time Turing reducible to $F_{\mathcal{C}}$; the theorem states that no such pair exists.

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Extended reading notes

Core claim

The paper's central claim is Corollary 4.13: if $\mathcal{C}^\zeta$ is a ribbon zesting of $\mathcal{C}$, then for any framed, oriented link $L$ colored by simple objects, the Reshetikhin-Turaev invariant $F_{\mathcal{C}^\zeta}(L)$ is polynomial-time Turing equivalent to $F_{\mathcal{C}}(L)$. Equivalently, zesting preserves the computational complexity of simply colored link invariants. The proof works by defining a local invariant $J_\zeta$ of $A$-shadow-colored tangles, showing that $F_{\mathcal{C}^\zeta}(T)$ is obtained from $F_{\mathcal{C}}(T)$ by interleaving it with $J_\zeta(T)$ through the warp product, and then giving a polynomial-time algorithm that evaluates $J_\zeta$ on links by converting a diagram to a braid closure and multiplying Boltzmann weights at crossings. In the pseudo-unitary setting $J_\zeta(L)$ coincides with a shadow rack cocycle invariant, so the correction factor is a discrete state-sum invariant rather than a generic quantum invariant.

Load-bearing premise

The polynomial-time equivalence assumes the zesting data are fixed algebraic numbers in a finite extension of $\mathbb{Q}$ known before the link diagram is given; if the data arrive as part of the input with unbounded size, or the ground field is not a number field, the proof's complexity bound no longer applies.

Editorial extensions

If this is right

If $\mathcal{C}$ and $\mathcal{C}^\zeta$ are related by zesting, any algorithm that computes $F_{\mathcal{C}}$ on simply colored links yields an algorithm for $F_{\mathcal{C}^\zeta}$ with only polynomial overhead, and conversely.
The zest invariant $J_\zeta(L)$ itself is polynomial-time computable from a link diagram, so the multiplicative correction never introduces super-polynomial work under the paper's assumptions.
For pseudo-unitary categories, zesting-related Reshetikhin-Turaev invariants differ by a shadow rack cocycle invariant, so the complexity of those rack state-sum invariants is bounded by the complexity of the parent theory.
For $A$-manifolds built from $A$-modular categories, the analogous $A$-manifold invariants are also polynomial-time Turing equivalent, extending the result beyond links to a class of $3$-manifold invariants.
Zesting equivalence classes give candidate equivalence classes for complexity-theoretic hierarchies of $(2+1)$-dimensional topological quantum field theories and topological phases.

Reading between the lines

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

The paper leaves open whether zesting preserves the complexity of $3$-manifold invariants without $A$-structure; because the surgery color mixes grading degrees, the same factorization does not directly apply, and the paper's own Remark 5.20 suggests the relative complexity there is genuinely subtler.
The shadow rack cocycle identification suggests that the correction factor, and hence the complexity-equivalence class, depends only on a cohomology class of the zesting data rather than on the full category, a structural reduction the paper does not pursue.
The framework suggests a testable extension: apply the local formula to families of finite-group gauge theories where the zesting data arise from group cohomology, to see whether the known hardness or easiness of coloring invariants is stable under the complexity-equivalence relation.
As an immediate corollary of the Turing equivalence, any hardness proof for the Reshetikhin-Turaev invariant of one category automatically transfers to every zesting of that category, which could be used to certify hardness of invariants in less-studied categories.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main result is real: zesting preserves the polynomial-time Turing complexity of Reshetikhin–Turaev invariants for links colored by simple objects. The new piece is the tangle-level invariant J_ζ, built as a 2-functor from shadow-colored tangles into the delooping of the pointed subcategory. That gives a local, diagrammatic computation, a polynomial-time algorithm, and the connection to shadow rack cocycle invariants. The rack cocycle identification is a nice byproduct and may have legs beyond this paper. The extension to A-manifold invariants is a sensible bonus, and the relative-complexity framing is stated honestly.

I read the main proofs as sound. The factorization theorem for tangles is proven by generators and relations, with invariance reduced to the ribbon property of C^ζ from Delaney–Galindo–Plavnik–Rowell–Zhang. The rack cocycle verification in Appendix B is direct. The reliance on earlier work — especially [Del+20] and [DKP25] — is substantial, but the new local formalism and the complexity reduction are not simply restatements; the circularity burden is low. The citation pattern is honest.

The soft spot is exactly the one the stress-test flags. Theorem 4.12 says "given ribbon zesting data ζ and a diagram D" the invariant is computable in polynomial time. But the proof assumes the values of ν, t, ϵ lie in a fixed finite extension K of Q precomputed independently of the diagram. That assumption is load-bearing: computing J_ζ by the local formula requires arithmetic in K as constant-time operations. If ζ is part of the input with unbounded degree or coefficient size, the stated complexity bound does not follow. Corollary 4.13 inherits the problem, since as written it ranges over arbitrary ribbon fusion categories over algebraically closed fields of characteristic 0, including non-computable fields. This is not a gap in the factorization math; it is an unstated hypothesis in the complexity statement. The fix is routine: state the theorem for fixed C and fixed ζ, or include ζ in the input size and prove the bound in that measure, and require the ground field to be a number field. The footnote typo about t(i,i)^{-1} should be corrected but is minor.

Who benefits: people working on complexity of TQFT invariants, topological quantum computation, and zesting as a structural operation. The paper deserves a serious referee; it is not a desk-reject. I would send it out, with the expectation that the authors clarify the complexity assumptions and fix the typo.

Referee Report

2 major / 5 minor

Summary. The paper studies the effect of ribbon zesting on the computational complexity of Reshetikhin-Turaev link invariants. The authors construct a 2-functor J_ζ from a 2-category of A-shadow-colored tangles to the delooping of the pointed subcategory of invertible objects, giving a local, diagrammatic definition of the zesting correction invariant J_ζ(T). They prove a factorization theorem F_{C^ζ}(T) = F_C(T) ⋊ J_ζ(T) for homogeneous tangles, extend this to links, and identify J_ζ(L) (up to a sign that vanishes in the pseudo-unitary case) with a shadow rack cocycle invariant. Using Vogel's algorithm to convert link diagrams to braid closures, they then claim polynomial-time computability of J_ζ(L) and, as a corollary, polynomial-time Turing equivalence of the link invariants for zesting-related categories. The framework is extended to define invariants of closed 3-manifolds with A-structure, and a similar complexity equivalence is claimed for A-modular categories.

Significance. The paper gives a clean, local formalism for the zesting correction invariant, complementing the earlier link-level results of [DKP25] with a tangle-level 2-functor construction. The identification of the zest invariant with a shadow rack cocycle invariant is a novel connection between zesting and rack cohomology that may be of independent interest. The complexity result, if properly qualified, is a useful step toward organizing TQFT invariants into complexity-theoretic hierarchies, and the paper explicitly relates it to existing work on quantum braid group representations under zesting. The proofs are largely detailed and the factorization argument is sound; Appendix B verifies the shadow rack cocycle condition directly from the zesting axioms.

major comments (2)

[Section 4.4.1, Theorem 4.12, Corollary 4.13] The proof of Theorem 4.12 assumes that the zesting data ν, t, ϵ take values in a fixed finite extension K of Q that is pre-computed independently of the diagram D or link L, but this hypothesis is not stated in the theorem and is omitted from Corollary 4.13. As stated, Corollary 4.13 quantifies over ribbon fusion categories over an algebraically closed field of characteristic 0, which need not be number fields, and for which 'polynomial-time Turing equivalence' is not canonically defined; if the zesting data ζ is instead given as part of the input with unbounded algebraic degree or coefficient bit-length, the running time measured only in n(D)+c(D) is insufficient to account for the cost of arithmetic in K. Because the complexity equivalence is obtained by multiplying or dividing by J_ζ(L), the efficient evaluability of J_ζ is load-bearing. I recommend restating Theorem 4.12 and Corollaries 4.13 and 5.19 for a fixed category C and fixed zesting data valued in a fixed number field, or explicitly including the bit-length of the zesting data in the input size and proving the bound in that measure.
[Section 5.3.1, Corollary 5.19] Corollary 5.19 is asserted to follow obviously from the argument of Theorem 4.12 and therefore inherits the same unstated fixed-field/number-field restriction on the zesting data and the ground field. In addition, the definition of the A-manifold invariant Z_ζ(M,ω) in equation (38) uses J_ζ(e,L,ω) on a surgery presentation, so a precise complexity statement requires explicit conventions for how the surgery link, its linking matrix, and the A-coloring are encoded. The current statement, 'ZC(M,ω) and ZCζ(M,ω) are polynomial-time Turing equivalent,' is not well-posed unless these inputs and the field are specified. Please state the precise input conventions and the hypotheses under which the equivalence holds.

minor comments (5)

[Section 3.3, Proposition 3.12] The displayed equality 'J_ζ(loop) = J_ζ(loop) = ϵ(a)' is ambiguous because the two sides refer to the counterclockwise and clockwise oriented loops, yet both are written with the same symbol; please use distinct notation and clarify that both values equal ϵ(a) because ϵ(a) = ϵ(a)^{-1}.
[Section 4.4.1] The phrase 'we can perform the multiplications in the formula for J(B) for free' is informal; since K is fixed, the intended meaning is 'at constant cost per field operation,' which should be stated explicitly.
[Section 4.2, equations (28)–(31)] The passage from the morphism-valued zesting conditions in Definition 2.4 to the scalar-valued equations (28)–(31) is somewhat compressed; a brief remark on how the braiding of the pointed subcategory is evaluated to obtain the β-term would help the reader verify the equations.
[Example 4.2] The cancellation of the ν terms and all but one of the t terms is stated without showing the intermediate algebra; adding a line or two of computation would make the example self-contained and easier to check.
[Definition 4.5, equation (32)] The shadow rack 2-cocycle condition is written with fractions, which makes the exponents hard to read; presenting it as a product of weights with explicit powers would improve readability.

Circularity Check

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No significant circularity: the zesting-factorization and the polynomial-time evaluation of J_zeta are proven from the zesting data by an independent local tangle functor, not assumed.

full rationale

The central claim, Corollary 4.13, rests on two components: the factorization F_{C^zeta}(L) = J_zeta(L) F_C(L) (Theorem 2.7 / Lemma 3.8) and the polynomial-time computability of J_zeta (Theorem 4.12). Neither reduces to its inputs by definition. J_zeta is defined independently as a 2-functor on A-shadow-colored tangles with explicit values on generators (equations (20)--(25)), and Theorem 3.7 proves diagram invariance using the standard tangle relations together with the established fact that zesting produces a ribbon category. The factorization is then derived, not posited, in Lemma 3.8 by checking the tangle generators. The computation of J_zeta on a link is a local product of the scalar weights phi_i(a,b) and epsilon-factors, so Theorem 4.12 is a genuine algorithm rather than a restatement of the conclusion. The paper relies heavily on [Del+20] and [DKP25], both involving the first author, but these are published peer-reviewed constructions, and the present proof does not merely cite the conclusion: it rebuilds the local formalism. One caveat is explicitly stated in the proof of Theorem 4.12 (Section 4.4.1): the zesting data values 'lie in a fixed finite extension K of Q which is pre-computed independently of D or L.' This is a load-bearing but unstated hypothesis about how zesting data are supplied as input, and it limits the scope of the polynomial-time claim when zeta is part of the input with unbounded algebraic complexity or when k is not a number field. That is a correctness/scope caveat, not a circular reduction: no parameter is fitted to the link invariant, and Corollary 4.13 is not equivalent to its assumptions by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted constants appear; the zesting data zeta is an input, not a fitted parameter. The axioms are the standard categorical background plus the specific assumption that zesting data values lie in a fixed number field for the complexity result. No new physical or algebraic entities are postulated.

assumptions (6)

domain assumption The zesting construction produces a ribbon fusion category C^zeta from C and zesting data zeta (Del+20).
Used throughout; Theorem 2.7 and the whole tangle formalism presuppose the existence and ribbon property of the zested category. See Section 2.2 and [Del+20].
standard math Reshetikhin-Turaev construction defines a ribbon functor from the category of framed tangles to C (RT90, RT91, Tur16).
Used to define FC and to justify the factorization and invariance checks in Sections 2.3, 3.3, and 5.2.
domain assumption The universal grading group A is finite, abelian, and the grading is faithful; homogeneous objects exist in every degree.
Used in Lemma 3.8 to choose a strand S of any degree and in the definition of A-colored tangles. For any fusion category its universal grading is faithful by [GN08, Theorem 3.5].
standard math The category C is strict (monoidally), or equivalent to a strict one by coherence.
The warp product and graphical calculus in Section 2.3 assume left-associated strict tensor products; strictification is standard by Mac Lane coherence.
domain assumption The zesting data values nu, t, epsilon lie in a fixed finite extension K of Q, precomputed independently of the link diagram.
Assumed in the proof of Theorem 4.12 to make field arithmetic polynomial-time; load-bearing for the complexity claims.
domain assumption The ground field k is algebraically closed of characteristic 0.
Stated in Section 2.1; needed for the scalar morphism <f> and for the classification results used in the zesting data.

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Pith. "Pith review of Zesting and the relative complexity of Reshetikhin-Turaev invariants." pith.science (2026). https://pith.science/paper/QPMDYWAL

@misc{pith2026260802795,
  author       = {Pith},
  title        = {Pith review of: Zesting and the relative complexity of Reshetikhin-Turaev invariants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QPMDYWAL}},
  note         = {Machine review of arXiv:2608.02795}
}

abstract

We show that the computational complexity of Reshetikhin-Turaev invariants of simply colored links is preserved when their underlying ribbon fusion categories are related by the zesting construction. Zesting modifies an $A$-graded ribbon fusion category $\mathcal{C}$ with additional algebraic data $\zeta$ to produce a new category $\mathcal{C}^{\zeta}$ whose link invariants are known to differ from those of $\mathcal{C}$ by an invariant of $A$-colored links $\mathcal{J}_{\zeta}$ depending only on $\zeta$. Building on this understanding and on earlier work on quantum braid group representations under zesting, our result suggests how zesting contributes to the organization of (2+1)D topological quantum field theories and topological phases into complexity-theoretic hierarchies. To prove our main result we develop a local formalism analogous to the Reshetikhin-Turaev construction to compute \emph{tangle} invariants $\mathcal{J}_{\zeta}(T)$, which leads to a polynomial time algorithm to compute invariants of links $\mathcal{J}_{\zeta}(L)$. A byproduct of our construction is an identification (up to a sign) of the link invariants $\mathcal{J}_{\zeta}(L)$ as rack cocycle invariants, which may be of independent interest. Our formalism also extends to define invariants of closed $3$-manifolds with $A$-structure and we obtain similar complexity results for homotopy quantum field theories built from $A$-modular fusion categories.

Figures

Figures reproduced from arXiv: 2608.02795 by the authors.

**Figure 1.** A shadow coloring satisfies these relations at every edge, depending on its orientation. Here the black label corresponds to the edge and the gold labels to the regions. 6 This is the usual name in the quandle literature [CKS01]. Shadow colorings of tangle diagrams are unrelated to Turaev’s shadows [Tur16, Chapter IX]. 7 Note that a ∗ = (a, −1) and (a−1 , 1) are not the same 1-morphism, although they correspond … view at source ↗

**Figure 2.** Colored, oriented, framed Reidemeister moves as relators [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗

**Figure 3.** Shadow colorings around a crossing. [IK14] A. Inoue and Y. Kabaya, “Quandle homology and complex volume”. arXiv doi [Caz22] N. Cazet, The shadow quandle cocycle invariant of knotoids. arXiv 12 When A is finite the set of A-colorings of L is also finite, so one can eliminate the dependence on the coloring by taking a sum or product of the Z(L) over all colorings. This approach is common in the literature, but in th… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: A negative blowup move, where the box labeled by [PITH_FULL_IMAGE:figures/full_fig_p032_4.png]

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Works this paper leans on

6 extracted references · 5 canonical work pages

[1]

On invariantsofmodularcategoriesbeyondmodulardata

[Bon+19] ParsaBonderson,ColleenDelaney,CésarGalindo,EricC.Rowell,AlanTran,etal.“On invariantsofmodularcategoriesbeyondmodulardata”.In: J. Pure Appl. Algebra223.9 (2019),pp.4065–4088. issn:0022-4049,1873-1376. doi: 10.1016/j.jpaa.2018.12

work page doi:10.1016/j.jpaa.2018.12 2019
[6]

Quandle homology and complex volume

[IK14] AyumuInoueandYuichiKabaya.“Quandlehomologyandcomplexvolume”.In: Ge- ometriae Dedicata171 (2014), pp. 265–292.issn: 0046-5755.doi: 10.1007/s10711- 013-9898-2.arXiv: 1012.2923 [math.GT]. [JVW90] F. Jaeger, D. L. Vertigan, and D. J. A. Welsh. “On the computational complexity of theJonesandTuttepolynomials”.In: Math. Proc. Cambridge Philos. Soc.108.1(1...

work page Pith review arXiv 2014
[10]

Quandle Cohomology and State-sum Invariants of Knotted Curves and Surfaces

4230/LIPIcs.TQC.2025.5. [Car+03] JScottCarter,DaneilJelsovsky,SeiichiKamada,LaurelLangford,andMasahicoSaito. “Quandlecohomology andstate-suminvariantsofknotted curvesand surfaces”.In: Transactions of the American Mathematical Society355.10(2003),pp.3947–3989.arXiv: math/9903135 [math.GT]. [Caz22] Nicholas Cazet.The shadow quandle cocycle invariant of knot...

work page Pith review arXiv 2003
[17]

TowardsaComplexity-TheoreticDichotomy for TQFT Invariants

[BS25] NicolasBridgesandEricSamperton.“TowardsaComplexity-TheoreticDichotomy for TQFT Invariants”. In:20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025).Ed.byBillFeﬀerman.Vol.350.Leibniz InternationalProceedingsinInformatics(LIPIcs).Dagstuhl,Germany:SchlossDagstuhl –Leibniz-ZentrumfürInformatik,2025,5:1–5:21. i...

work page 2025
[2005]

Zesting produces modular isotopes andexplainstheirtopologicalinvariants

05544 [math.QA]. [DKP25] Colleen Delaney, Sung Kim, and Julia Plavnik. “Zesting produces modular isotopes andexplainstheirtopologicalinvariants”.In: Quantum TopologyPublishedonlineﬁrst (2025).doi: 10.4171/qt/238.arXiv: 2107.11374 [math.QA]. [DMS25] ColleenDelaney,ClémentMaria,andEricSamperton.“AnAlgorithmforTambara- YamagamiQuantumInvariantsof3-Manifolds,...

arXiv 2025
[2022]

The shadow quandle cocycle invariant of knotoids

arXiv: 2207.02330 [math.GT]. [CFW16] ShawnX.Cui,MichaelH.Freedman,andZhenghanWang.“Complexityclassesasmath- ematicalaxiomsII”.In: Quantum Topol.7.1(2016),pp.185–201. issn:1663-487X,1664- 073X.doi: 10.4171/QT/75. [CKS01] J. Scott Carter, Seiichi Kamada, and Masahico Saito. “Geometric interpretations of quandle homology”. In:J. Knot Theory Ramiﬁcations10.3 ...

work page Pith review arXiv 2016

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

On invariantsofmodularcategoriesbeyondmodulardata

[Bon+19] ParsaBonderson,ColleenDelaney,CésarGalindo,EricC.Rowell,AlanTran,etal.“On invariantsofmodularcategoriesbeyondmodulardata”.In: J. Pure Appl. Algebra223.9 (2019),pp.4065–4088. issn:0022-4049,1873-1376. doi: 10.1016/j.jpaa.2018.12

work page doi:10.1016/j.jpaa.2018.12 2019

[2] [6]

Quandle homology and complex volume

[IK14] AyumuInoueandYuichiKabaya.“Quandlehomologyandcomplexvolume”.In: Ge- ometriae Dedicata171 (2014), pp. 265–292.issn: 0046-5755.doi: 10.1007/s10711- 013-9898-2.arXiv: 1012.2923 [math.GT]. [JVW90] F. Jaeger, D. L. Vertigan, and D. J. A. Welsh. “On the computational complexity of theJonesandTuttepolynomials”.In: Math. Proc. Cambridge Philos. Soc.108.1(1...

work page Pith review arXiv 2014

[3] [10]

Quandle Cohomology and State-sum Invariants of Knotted Curves and Surfaces

4230/LIPIcs.TQC.2025.5. [Car+03] JScottCarter,DaneilJelsovsky,SeiichiKamada,LaurelLangford,andMasahicoSaito. “Quandlecohomology andstate-suminvariantsofknotted curvesand surfaces”.In: Transactions of the American Mathematical Society355.10(2003),pp.3947–3989.arXiv: math/9903135 [math.GT]. [Caz22] Nicholas Cazet.The shadow quandle cocycle invariant of knot...

work page Pith review arXiv 2003

[4] [17]

TowardsaComplexity-TheoreticDichotomy for TQFT Invariants

[BS25] NicolasBridgesandEricSamperton.“TowardsaComplexity-TheoreticDichotomy for TQFT Invariants”. In:20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025).Ed.byBillFeﬀerman.Vol.350.Leibniz InternationalProceedingsinInformatics(LIPIcs).Dagstuhl,Germany:SchlossDagstuhl –Leibniz-ZentrumfürInformatik,2025,5:1–5:21. i...

work page 2025

[5] [2005]

Zesting produces modular isotopes andexplainstheirtopologicalinvariants

05544 [math.QA]. [DKP25] Colleen Delaney, Sung Kim, and Julia Plavnik. “Zesting produces modular isotopes andexplainstheirtopologicalinvariants”.In: Quantum TopologyPublishedonlineﬁrst (2025).doi: 10.4171/qt/238.arXiv: 2107.11374 [math.QA]. [DMS25] ColleenDelaney,ClémentMaria,andEricSamperton.“AnAlgorithmforTambara- YamagamiQuantumInvariantsof3-Manifolds,...

arXiv 2025

[6] [2022]

The shadow quandle cocycle invariant of knotoids

arXiv: 2207.02330 [math.GT]. [CFW16] ShawnX.Cui,MichaelH.Freedman,andZhenghanWang.“Complexityclassesasmath- ematicalaxiomsII”.In: Quantum Topol.7.1(2016),pp.185–201. issn:1663-487X,1664- 073X.doi: 10.4171/QT/75. [CKS01] J. Scott Carter, Seiichi Kamada, and Masahico Saito. “Geometric interpretations of quandle homology”. In:J. Knot Theory Ramiﬁcations10.3 ...

work page Pith review arXiv 2016