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Fundamental generalized Legendrian rack and classical invariants

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Pith's one-line read Two Legendrian knots with isomorphic fundamental GL-racks must have the same classical invariants, or both reversed.

desk verdict A genuinely new partial converse to Kimura, with the right proof idea, but the proof as written omits cusp relations and a load-bearing case analysis; send to a referee. read the letter →

arxiv 2507.18500 v1 pith:VYA7V4JR submitted 2025-07-24 math.GT

Zhiyun Cheng , Zhiyi He This is my paper · ORCID
classification math.GT MSC 57K12
keywords LegendrianknotfundamentalGL-rackThurston-Bennequinnumberrotationrackinvariantfrontprojectionclassicalinvariants
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 aims to establish that the fundamental generalized Legendrian rack (GL-rack), an algebraic invariant built from the front diagram of a Legendrian knot, determines the knot's two classical numerical invariants up to a simultaneous sign. Concretely, the theorem states that if two Legendrian knots have isomorphic fundamental GL-racks, then either their Thurston-Bennequin numbers and rotation numbers are equal, or both are the negatives of the other's. This matters because it shows the GL-rack carries real geometric information rather than being a purely combinatorial decoration, and it narrows the search for invariants that the GL-rack could fail to distinguish. The proof extracts the classical data from signed exponent sums in the rack's cyclic presentation.

What carries the argument

The load-bearing object is the fundamental GL-rack $\mathrm{GLR}_K$ of a Legendrian knot, presented from a front diagram as a free GL-rack on the arcs modulo crossing and cusp relations, and specifically its cyclic presentation with relations $r_i: u^{p_i}d^{q_i}(x_i) *^{\epsilon_i} x_{k_i} = x_{i+1}$ for $1 \le i \le m$ (indices modulo $m$). From these relations the proof forms the three total sums $p = \sum_i p_i$, $q = \sum_i q_i$, and $\omega = \sum_i \epsilon_i$, which are related to the writhe and cusp counts, hence to the Thurston-Bennequin and rotation numbers. The argument then uses permutation GL-racks on $\mathbb{Z}_k$, where $a * b = \sigma(a)$ and $ud = \sigma^{-1}$, choosing three different pairs $(u,d)$ — $(\sigma^{-1}, \mathrm{id})$, $(\mathrm{id}, \sigma^{-1})$, and $(\sigma^{-2}, \sigma)$ — to force the absolute-value equalities $|\omega-p|=|\tilde{\omega}-\tilde{p}|$, $|\omega-q|=|\tilde{\omega}-\tilde{q}|$, and $|\omega-2p+q|=|\tilde{\omega}-2\tilde{p}+\tilde{q}|$. The final step, meant to convert these three absolute equalities into the two sign patterns, is the part of the machinery that carries the theorem's conclusion.

What would settle it

Enumerate all sign patterns for the three pairs $(\omega-p, \tilde{\omega}-\tilde{p})$, $(\omega-q, \tilde{\omega}-\tilde{q})$, and $(\omega-2p+q, \tilde{\omega}-2\tilde{p}+\tilde{q})$ that satisfy the three absolute-value equalities; if any pattern other than all-aligned or all-reversed survives, the proof's final inference is broken. Then search for two Legendrian knots realizing such sums with isomorphic fundamental GL-racks but unequal, non-opposite $(\mathrm{tb}, \mathrm{rot})$ — such a pair would refute Theorem 1.1 directly.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is Theorem 1.1: for any two Legendrian knots $K_1$ and $K_2$ with isomorphic fundamental GL-racks, the pair $(\mathrm{tb}(K_1), \mathrm{rot}(K_1))$ is either equal to $(\mathrm{tb}(K_2), \mathrm{rot}(K_2))$ or equal to $(-\mathrm{tb}(K_2), -\mathrm{rot}(K_2))$. The argument packages the front diagram into a cyclic presentation of the fundamental GL-rack with relations $u^{p_i}d^{q_i}(x_i) *^{\epsilon_i} x_{k_i} = x_{i+1}$, then forms the total sums $p$, $q$, and $\omega$ of the exponents and crossing signs. Evaluating the fundamental GL-rack in permutation GL-racks on cyclic sets with three carefully chosen pairs $(u,d)$ yields absolute-value equalities for $\omega - p$, $\omega - q$, and $\omega - 2p + q$. The paper asserts that these three equalities leave only the two sign patterns in the theorem; it also records a slice-knot corollary and asks whether the converse holds.

Load-bearing premise

The theorem stands on two unstated premises: that every front diagram yields a cyclic presentation of the fundamental GL-rack with exactly $m$ relations of the form $u^{p_i}d^{q_i}(x_i) *^{\epsilon_i} x_{k_i} = x_{i+1}$, and that three absolute-value equalities force only the two sign patterns; the latter is asserted without a full case analysis.

Editorial extensions

If this is right

  • If two Legendrian knots with the same underlying topological type have isomorphic fundamental GL-racks, their classical invariants cannot be unrelated: the only freedoms are simultaneous equality or simultaneous sign flip.
  • For slice knot types, isomorphic fundamental GL-racks imply the Thurston-Bennequin and rotation numbers are equal, since the slice-genus bound forces both Thurston-Bennequin numbers to be negative.
  • Any invariant that factors through the fundamental GL-rack must take equal values on Legendrian knots whose $(\mathrm{tb}, \mathrm{rot})$ are either equal or both opposite.
  • The paper's Question 3.2 — whether two Legendrian knots with the same classical invariants can have distinct fundamental GL-racks — becomes the natural completeness test for the invariant.
  • The two knots in Example 2.13 show the fundamental GL-rack is not a complete Legendrian invariant, because they share it while having equal classical invariants.

Reading between the lines

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

  • The proof's reliance on cyclic presentations suggests the three absolute-value equalities are the entire content of the invariant's classical information; extending the same permutation-rack trick to other rack-valued invariants could yield analogous constraints for those invariants.
  • If the asserted 'only two possibilities' step is completed, the theorem would imply that the absolute values $|\mathrm{tb}|$ and $|\mathrm{rot}|$ are invariant under GL-rack isomorphism; one could then use classical knot tables to identify candidate Legendrian knots that might share a fundamental GL-rack.
  • A testable extension: for a fixed topological knot type, classify all possible sums $(p,q,\omega)$ arising from front diagrams and check whether any two distinct Legendrian representatives realize a mixed sign pattern of the three equalities; finding one would expose a counterexample to the proof's final step.
  • The two knots in Example 2.13 suggest that isomorphism of fundamental GL-racks may be common among Legendrian knots sharing classical invariants, making Question 3.2's converse especially delicate.
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Editorial analysis

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

Referee Report

2 major / 3 minor

Summary. This paper studies the fundamental generalized Legendrian rack (GL-rack) of a Legendrian knot, a rack equipped with two cusp automorphisms u and d, following Kimura and Karmakar--Saraf--Singh. The main result, Theorem 1.1, asserts that if two Legendrian knots K1 and K2 have isomorphic fundamental GL-racks, then either their Thurston--Bennequin numbers and rotation numbers agree, or both are negated. The proof colors front diagrams by finite cyclic permutation GL-racks and derives three absolute-value identities involving the total numbers of up cusps, down cusps, and the writhe, then concludes that only two sign patterns are possible. A corollary states that if one of the knots is slice, the classical invariants are equal. Two examples illustrate that GL-rack colorings can distinguish some Legendrian knots and that isomorphic GL-racks need not imply Legendrian isotopy.

Significance. If Theorem 1.1 is correct, it is a significant partial converse to Kimura's theorem: the fundamental GL-rack, an algebraic invariant of the front, almost determines the two classical Legendrian invariants, fixing them up to a simultaneous sign change. The proof strategy is appealing and elementary, and the three-family permutation-rack argument is a nice tool. The examples are informative, showing both strength (Example 2.12) and limitation (Example 2.13) of the invariant. The main obstacle is that the proof relies on a presentation of the fundamental GL-rack that is not justified and is false for cusp-only fronts; this must be repaired before the theorem can be accepted.

major comments (2)
  1. [Section 3, Proof of Theorem 1.1, first paragraph] The proof assumes without proof that GLR_K admits a presentation <x_1,...,x_m | r_1,...,r_m> in which every r_i is a crossing relation of the form u^{p_i}d^{q_i}(x_i) *^{\epsilon_i} x_{k_i} = x_{i+1}. This contradicts Definition 2.10 and Figure 4, where cusp relations u(x)=y and d(x)=y are part of the presentation and arcs are cut at cusps as well as undercrossings. For the standard two-cusp unknot front there are no crossing relations at all, so the asserted presentation is not true for that front. The authors need a lemma that chooses a front with at least one crossing, labels the arcs immediately after undercrossings, eliminates cusp generators using the cusp relations, and proves that p=\sum p_i, q=\sum q_i, and \omega=\sum \epsilon_i equal the numbers of up cusps, down cusps, and the writhe; the no-crossing case must be treated separately. As written, the proof of Theorem 1.1 depends on unproved and, for cusp-only fronts, false identities.
  2. [Section 3, paragraph beginning "Based on these"] The step from the three absolute-value equalities |\omega-p|=|\tilde{\omega}-\tilde{p}|, |\omega-q|=|\tilde{\omega}-\tilde{q}|, and |\omega-2p+q|=|\tilde{\omega}-2\tilde{p}+\tilde{q}| to the dichotomy (\omega-p=\tilde{\omega}-\tilde{p} and \omega-q=\tilde{\omega}-\tilde{q}) or (\omega-p+\tilde{\omega}-\tilde{p}=0 and \omega-q+\tilde{\omega}-\tilde{q}=0) is asserted without proof. The underlying algebraic lemma is true: one checks the cases B=\pm A and D=\pm C in the third equality. However, because this inference is load-bearing for the theorem, it should be stated and proved explicitly. This is a fillable gap rather than a false claim.
minor comments (3)
  1. [Example 2.13] In the displayed data for the Chekanov--Eliashberg example, 'rot(K3)=rot(K3)=0' should presumably read 'rot(K3)=rot(K4)=0', and in the definition of f, 'f(x6)=d^2(x6)' should be 'f(x6)=d^2(y6)' because f maps into GLR_{K4}.
  2. [Corollary 3.1] The sentence 'the knot quandle can be recovered from the generalized GL-rack by setting u=d=id' is informal; setting the cusp automorphisms to the identity is not a recovery procedure on a fixed GL-rack. Please phrase this as a quotient or as forgetting the cusp relations, and justify that the result is the knot quandle.
  3. [Example 2.12] The claimed conclusion Col_{Z_9}(K_1)=0 would be easier to verify if the computation leading to \sigma^{-7}(\psi(x_1))=\psi(x_1) were displayed rather than summarized by an ellipsis.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof derives constraints on the classical invariants from independent GL-rack coloring counts.

full rationale

The derivation chain of Theorem 1.1 is not circular. The paper defines the fundamental GL-rack GLR_K from a front diagram (Definition 2.10 and Figure 4), imports the Reidemeister invariance theorem from prior work (Theorem 2.11, citing [14]), and then proves the main implication by counting homomorphisms into permutation GL-racks over Z_k. The colorability constraints phi(x1) = sigma^(omega-p)(phi(x1)) etc. are consequences of the rack relations, and the absolute-value equalities are inferred by comparing existence of homomorphisms; they are not assumed. The final translation into statements about tb and rot uses the standard front-diagram formulas for these invariants, not a fitted parameter or a renaming. The cited prior results about GL-racks are external inputs, not self-citations that carry the main claim, and the paper does not define GLR_K in terms of tb or rot. A reader may object that the proof's claim that GLR_K has a presentation using only crossing relations of the stated form is not justified for fronts with cusp-only arcs, and that the final two-case inference is asserted rather than fully derived; however, those are correctness gaps, not circular reductions. No parameter is fitted and no conclusion is assumed, so the argument is self-contained in the sense relevant to circularity.

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

The ledger is small because the paper is a theorem proof rather than a model-building exercise. The main external inputs are the cited invariance theorem for GL-racks and the standard rack/quandle framework. The only ad hoc item is the unproved algebraic dichotomy at the end of the proof, which is internal to the argument.

assumptions (4)
  • domain assumption The fundamental GL-rack GLR_K is invariant under Legendrian Reidemeister moves (Theorem 2.11, cited to [14]).
    The main theorem compares isomorphism classes of GLR_K and GLR_K', so the invariant must be well-defined on Legendrian isotopy classes; the paper cites [14] rather than proving this.
  • domain assumption The rack and GL-rack axioms (Definition 2.5) are the correct algebraic mirror of Legendrian front-projection moves.
    All claims inherit the definitions from [14,15,4]; no independent justification is given beyond the cited framework.
  • ad hoc to paper The algebraic lemma that |A|=|B|, |C|=|D| and |2A-C|=|2B-D| forces either (A=B and C=D) or (A=-B and C=-D).
    This is the unproved case split in Section 3, paragraph 'Based on these, there are only two possibilities'. It is true but not demonstrated, and it is needed to finish the proof.
  • standard math Joyce-Matveev quandle classification and Bennequin's inequality are used in Corollary 3.1.
    Well-known theorems cited as [13,17] and stated before the corollary; they support the slice-knot corollary, not the main theorem.

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Pith. "Pith review of Fundamental generalized Legendrian rack and classical invariants." pith.science (2026). https://pith.science/paper/VYA7V4JR

@misc{pith2026250718500,
  author       = {Pith},
  title        = {Pith review of: Fundamental generalized Legendrian rack and classical invariants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VYA7V4JR}},
  note         = {Machine review of arXiv:2507.18500}
}
read the original abstract

In this paper, we prove that if two Legendrian knots have isomorphic fundamental GL-racks, then either they have the same Thurston-Bennequin number and the same rotation number, or they have the opposite Thurston-Bennequin numbers and opposite rotation numbers.

Figures

Figures reproduced from arXiv: 2507.18500 by the authors.

Figure 1
Figure 1. cusps and crossing [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Legendrian Reidemeister moves In order to distinguish different Legendrian knots, many invariants have been studied in the literature. Here we briefly review the classical invariants of Legendrian knots. The reader is referred to [10] for more details and recent progress. • Topological knot type. Obviously, the underlying topological knot type of a Legen￾drian knot K is an invariant, which is usually denoted by k(K)… view at source ↗
Figure 3
Figure 3. Stabilization 2.2. Fundamental GL-rack. Recall that a rack is a pair (X, ∗) where X is a set and ∗ is a binary operation satisfying the following axioms: (1) the map sx : X → X, where sx(y) = y ∗ x is a bijection for all x ∈ X; (2) (x ∗ y) ∗ z = (x ∗ z) ∗ (y ∗ z) for all x, y, z ∈ X. A quandle is a rack satisfying that x ∗ x = x for all x ∈ X. These three axioms correspond to the Reidemeister moves in knot theory. E… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Then, the free GL-rack (F GLR(D(K)), ∗, u, d) associated to the front diagram D(K) is called the fundamental GL-rack of K, denoted by GLRK. x x x u(x) d(x) y x ∗ y [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Two Legendrian knots with topological knot type 51 and the fundamental GL-rack GLRK2 is generated by {y1, y2, y3, y4, y5, y6, y7, y8} with relations:    u(y1) ∗ −1 y7 = y2 y2 ∗ −1 y4 = y3 d(y3) ∗ −1 y7 = y4 y4 ∗ −1 y1 = y5 d(y5) ∗ −1 y7 = y…
Figure 6
Figure 6. Figure 6: Chekanov-Eliashberg knots Remark 2.14. A natural idea to promote GL-rack is to distinguish between the left up (dwon) cusp and the right up (down) cusp. Following this idea, recently Kimura introduced the notion of fundamental 4-Legendrian rack. However, the fundamenta…
Figure 7
Figure 7. Figure 7: Reading fundamental GL-rack from front projection Proof. Consider a front projection of K1 and K2, respectively, say D1 and D2. We assume that the arc collection of D1 is {x1, x2, . . . , xm}, the arc collection of D2 is {y1, y2, . . . , yn}. Then their fundamental GL-…

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Works this paper leans on

20 extracted references · 15 canonical work pages · cited by 1 Pith paper

  1. [1]

    From racks to pointed Hopf algebras.Adv

    Nicol´ as Andruskiewitsch and Mat´ ıas Gra˜ na. From racks to pointed Hopf algebras.Adv. Math., 178(2):177– 243, 2003

    2003

  2. [2]

    Scott Carter, Alissa S

    J. Scott Carter, Alissa S. Crans, Mohamed Elhamdadi, Enver Karadayi, and Masahico Saito. Cohomology of Frobenius algebras and the Yang-Baxter equation. Commun. Contemp. Math. , 10:791–814, 2008

    work page 2008

  3. [3]

    Scott Carter, Daniel Jelsovsky, Seiichi Kamada, Laurel Langford, and Masahico Saito

    J. Scott Carter, Daniel Jelsovsky, Seiichi Kamada, Laurel Langford, and Masahico Saito. Quandle cohomology and state-sum invariants of knotted curves and surfaces. Trans. Amer. Math. Soc., 355(10):3947–3989, 2003

    work page 2003

  4. [4]

    Legendrian rack invariants of Legendrian knots

    Jose Ceniceros, Mohamed Elhamdadi, and Sam Nelson. Legendrian rack invariants of Legendrian knots. Commun. Korean Math. Soc. , 36(3):623–639, 2021

    work page 2021

  5. [5]

    Differential algebra of Legendrian links

    Yuri Chekanov. Differential algebra of Legendrian links. Invent. Math., 150(3):441–483, 2002

    2002

  6. [6]

    A survey of racks and quandles: some recent developments

    Mohamed Elhamdadi. A survey of racks and quandles: some recent developments. Algebra Colloq., 27(3):509– 522, 2020

    work page 2020

  7. [7]

    Quandles—an introduction to the algebra of knots , volume 74 of Student Mathematical Library

    Mohamed Elhamdadi and Sam Nelson. Quandles—an introduction to the algebra of knots , volume 74 of Student Mathematical Library. American Mathematical Society, Providence, RI, 2015

    work page 2015

  8. [8]

    Classification of topologically trivial Legendrian knots

    Yakov Eliashberg and Maia Fraser. Classification of topologically trivial Legendrian knots. In Geometry, topology, and dynamics (Montreal, PQ, 1995) , volume 15 of CRM Proc. Lecture Notes, pages 17–51. Amer. Math. Soc., Providence, RI, 1998

    work page 1995

Show all 20 references
  1. [9]

    Topologically trivial Legendrian knots.J

    Yakov Eliashberg and Maia Fraser. Topologically trivial Legendrian knots.J. Symplectic Geom., 7(2):77–127, 2009. FUNDAMENTAL GENERALIZED LEGENDRIAN RACK AND CLASSICAL INV ARIANTS 11

    2009

  2. [10]

    John B. Etnyre. Legendrian and transversal knots. In Handbook of knot theory , pages 105–185. Elsevier B. V., Amsterdam, 2005

    2005

  3. [11]

    Etnyre and Ko Honda

    John B. Etnyre and Ko Honda. Knots and contact geometry. I. Torus knots and the figure eight knot. J. Symplectic Geom., 1(1):63–120, 2001

    2001

  4. [12]

    Racks and links in codimension two

    Roger Fenn and Colin Rourke. Racks and links in codimension two. J. Knot Theory Ramifications, 1(4):343– 406, 1992

    1992

  5. [13]

    A classifying invariant of knots, the knot quandle

    David Joyce. A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra , 23(1):37–65, 1982

    1982

  6. [14]

    Generalised legendrian racks of legendrian links

    Biswadeep Karmakar, Deepanshi Saraf, and Mahender Singh. Generalised legendrian racks of legendrian links. arXiv:2301.06854, 2024

    2024

  7. [15]

    Bi-Legendrian rack colorings of Legendrian knots

    Naoki Kimura. Bi-Legendrian rack colorings of Legendrian knots. J. Knot Theory Ramifications, 32(4):Paper No. 2350029, 16, 2023

    2023

  8. [16]

    Dheeraj Kulkarni and T. V. H. Prathamesh. On rack invariants of legendrian knots. arXiv:1706.07626, 2017

    2017 arXiv

  9. [17]

    S. V. Matveev. Distributive groupoids in knot theory. Mat. Sb. (N.S.) , 119(161)(1):78–88, 160, 1982

    1982

  10. [18]

    Rubinsztein

    Ryszard L. Rubinsztein. Topological quandles and invariants of links. J. Knot Theory Ramifications , 16(6):789–808, 2007

    2007

  11. [19]

    Classification and structure of generalized legendrian racks

    Luc Ta. Classification and structure of generalized legendrian racks. arXiv:2504.12671, 2025

    2025 arXiv

  12. [20]

    On relations and homology of the Dehn quandle

    Joel Zablow. On relations and homology of the Dehn quandle. Algebr. Geom. Topol., 8(1):19–51, 2008. School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, MOE, Beijing Normal University, Beijing 100875, China Email address: czy@bnu.edu.cn School of Mat...

    2008

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