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The HOMFLY-PT polynomial and HZ factorisation

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Pith's one-line read For infinite families of hyperbolic knots generated by full twists and Jucys–Murphy braids, the paper proves that the HOMFLY–PT polynomial is determined by the Dubrovnik Kauffman polynomial, and conjectures this equivalence is exactly the…

desk verdict A useful extension of HZ factorisation to new hyperbolic families, with a HOMFLY–Kauffman theorem that is clearly stated but under-proved in the write-up. read the letter →

arxiv 2412.04933 v4 pith:6HTMQM2K submitted 2024-12-06 math-ph math.MP

Andreani Petrou , Shinobu Hikami This is my paper · ORCID
classification math-phmath.MP MSC 57K1057K1457K18
keywords HOMFLY-PTpolynomialHarer-ZagiertransformHZfactorisationKauffmanhyperbolicknotsJucys-MurphybraidKhovanovhomologyBPSinvariants
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

The paper studies the Harer–Zagier (HZ) transform, a discrete Laplace transform that sends the unnormalised HOMFLY–PT polynomial of a knot or link to a rational function in two parameters. For a special class of knots the transform factorises completely, and the paper shows this factorisation is preserved by two braid operations—full twists and concatenation with a Jucys–Murphy braid—so that infinite hyperbolic families can be built from a few seed knots. For the families $P(2,-3,\pm(2j+1))$, $K_{j,2}$, $K_{j,3}$ and $5_2^{3k}$, the paper proves that the HOMFLY–PT polynomial is fully encoded by the Dubrovnik Kauffman polynomial through the identity $H(K;a,z)=dKF(K;a,z)$, a relation previously known only for torus knots. The paper also shows this identity is equivalent to the vanishing of the two-crosscap BPS invariants of topological string theory and conjectures that the identity is exactly the criterion for HZ factorisability. A sympathetic reader should care because, if the conjecture holds, the complicated HOMFLY–PT polynomial of any HZ-factorisable knot is captured by a short list of integers that also point directly into the Khovanov homology table.

What carries the argument

The load-bearing object is the HZ transform and its inverse. The HZ transform is the formal series $Z(K;\lambda,q)=\sum_{N\ge0}\bar H(K;q^N,q)\lambda^N$, which replaces each monomial $a^\beta$ in the unnormalised HOMFLY–PT polynomial by $(1-\lambda q^\beta)^{-1}$. When the resulting rational function is a ratio of products of such simple factors, the knot is HZ-factorisable; the integers $m$, $\alpha_i$ and $e$ in (7) are the data that encode the whole family. The inverse transform (Theorem 3.1) is a contour integral around $\lambda=0$, and its residue evaluation yields formula (41) for the HOMFLY–PT polynomial and formula (51) for the Alexander polynomial in the factorised case. For the Kauffman side, the paper uses skein recursions (63)–(72) for the Dubrovnik Kauffman polynomial and shows, after cancellation of the extra terms, that $dKF$ satisfies exactly the same recursion as the HOMFLY–PT polynomial; induction on the seed torus knots completes the proof of Theorem 4.1. Proposition 5.1 is powered by the evaluation of the residue at $\lambda=0$ that gives the unnormalised Jones polynomial as $-\sum_i q^{\alpha_i}+\sum_j q^{\beta_j}$.

What would settle it

Take the next member of one of the families beyond the range used in the paper—for instance $P(2,-3,5)_3^k$ or $5_2^{3k}$ at a larger $k$—and compute its HZ transform symbolically. If the exponents depart from (10) or (9), or if the identity $H(K)=dKF(K)$ fails for that member, the family-level claim collapses; alternatively, a single knot with factorised HZ transform but $H\ne dKF$ would disprove Conjecture 4.1.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that HZ factorisability is not an accident of torus knots: it extends to hyperbolic knots and links and is generated by two geometric operations. The HZ transform of a factorisable knot has the form $Z(K;\lambda,q)=\lambda \prod_{i=0}^{m-2}(1-\lambda q^{\alpha_i})/\prod_{i=0}^{m}(1-\lambda q^{e+2i})$, and the paper derives a closed inverse-transform formula (41) that recovers the HOMFLY–PT polynomial from the integer data $(m,\{\alpha_i\},e)$. The main theorem (Theorem 4.1) proves for the families above that $H(K;a,z)=dKF(K;a,z):=gKF_{\mathrm{even}}(K;a,z)-\frac{z}{a-a^{-1}}gKF_{\mathrm{odd}}(K;a,z)$, where the two summands split the Dubrovnik Kauffman polynomial by parity in $z$. By equation (99) this equality is exactly the vanishing of the two-crosscap BPS invariants $\widehat N^{c=2}_{g,Q}$, and the paper verifies the equality for every HZ-factorisable knot up to 12 crossings, leading to the conjecture that the relation holds if and only if the HZ transform factorises. A separate structural result (Proposition 5.1) identifies the HZ exponents with the rows of the Khovanov table: for factorisable knots the graded Euler characteristic is $-\sum_{i=0}^{m-2}q^{\alpha_i}+\sum_{i=0}^{m}q^{\beta_i}$, so the Jones polynomial alone carries the full HOMFLY–PT information in these cases.

Load-bearing premise

The infinite-family HZ formulas, such as (9)–(25), are extrapolated from the first few computed members of each family, not proved for all powers of the twist; if any pattern breaks at a later member, that family is not HZ-factorisable and Theorem 4.1 cannot be applied to it.

Editorial extensions

If this is right

  • For the four infinite families in Theorem 4.1, the HOMFLY–PT polynomial can be computed from the Kauffman polynomial alone, and the two-crosscap BPS invariants $\widehat N^{c=2}_{g,Q}$ vanish.
  • Every member of any HZ-factorisable family generated by $F_m$ or $E_m$ has its HOMFLY–PT polynomial encoded by the small integer set $(m,\{\alpha_i\},e)$ through formula (41), so the whole family can be handled without computing large polynomials.
  • The Alexander polynomial of these factorisable families is given by the closed expression (51), and the paper records explicit formulas such as (52)–(56) for the pretzel, $5_2^{3k}$, and $10_{128}^{4k}$ families.
  • When HZ factorisation holds, the graded Euler characteristic of Khovanov homology is $-\sum_i q^{\alpha_i}+\sum_j q^{\beta_j}$; hence the Jones polynomial determines the HOMFLY–PT polynomial for these knots.
  • If Conjecture 4.1 is correct, checking HZ factorisability reduces to checking the single identity $H(K)=dKF(K)$, which is a finite algebraic computation from the two known polynomials.

Reading between the lines

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

  • A practical reading of Conjecture 4.1: one could test factorisability of an arbitrary knot by computing its Kauffman polynomial and comparing $dKF$ with $H$, avoiding the nontrivial construction of the HZ transform; the exhaustive up-to-12-crossing check in the paper is consistent with this shortcut.
  • The link story is less tidy than the knot story. The factorisable two-component link $L10n42$ violates the analogous relation (85), so if Conjecture 4.2 is right, factorisability for links implies (85) but not conversely; the exceptional link points to an extra invariant that the authors do not identify.
  • The observed 'Z2-lego' pieces in the Khovanov tables of consecutive factorisable knots suggest that each full twist adds one torsion block at a predictable position; if this pattern is proved, it would predict the Khovanov torsion of all members of the families from the HZ exponents alone.
  • The $q=1$ identity $Z(L;\lambda,1)=\sum_N N^l\lambda^N$ suggests that HZ factorisability imposes linear constraints on the exponents; testing these sum rules on randomly generated knots could provide a cheap numerical sieve for new factorisable families.
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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 / 5 minor

Summary. The paper studies the Harer-Zagier (HZ) transform of the HOMFLY-PT polynomial and identifies infinite families of hyperbolic knots and two-component links whose HZ transform takes a factorised form. It derives contour-integral formulas for inverting the HZ transform (Theorems 3.1 and 3.2), gives a closed expression (41) for HOMFLY-PT in the factorised case, and proves (or claims to prove) that for several HZ-factorisable families the HOMFLY-PT polynomial equals a specific combination dKF of the Dubrovnik Kauffman polynomial (Theorem 4.1). The paper further proposes that this HOMFLY-Kauffman relation is equivalent to the vanishing of two-crosscap BPS invariants, and connects the HZ exponents to Khovanov homology. The central conjecture is that HZ factorisation for knots is equivalent to the HOMFLY-Kauffman relation (Conjecture 4.1).

Significance. If the main results hold, the paper establishes a new structural relation between HOMFLY-PT and Kauffman polynomials for hyperbolic knot families, a conjectural criterion for HZ factorisability, and a physically interesting equivalence with vanishing two-crosscap BPS invariants. The manuscript contains genuinely useful components: the contour-inversion theorems are clean and correct, formula (41) is a compact and applicable closed form, Proposition 2.1 has a simple proof, and the paper provides extensive computational data and a sharply formulated conjecture. The main limitations are that the infinite-family HZ formulas are explicitly extrapolated from finitely many examples (footnote 3 and similar notes), and that the proof of Theorem 4.1 contains substantial unstated algebra and an unproven negative-j case. These gaps are load-bearing for the claims as stated.

major comments (3)
  1. [Sec. 2, footnote 3, Eqs. (9)-(25)] The infinite-family HZ transforms in Sec. 2 are not proven. Footnote 3 states that formulas such as (9)-(25) "are determined in this way, by extrapolating on the pattern exhibited by their first few members", with a rigorous derivation deferred to [22]. Since the paper's central claims concern infinite HZ-factorisable families, and Theorem 4.1 explicitly applies to these families, the hypotheses of that theorem and the associated tables of HZ parameters are not established for all members. Please either supply the missing derivations or clearly restate these infinite-family formulas as conjectural, and adjust Theorem 4.1 and the surrounding conclusions accordingly.
  2. [Sec. 4.1, Eqs. (63)-(73), proof of Theorem 4.1] The proof of Theorem 4.1 rests on unproved assertions. The Kauffman recursions (63), (65), (68), (70) and (72) are stated without derivation from the skein relation (58), and the reduction to the HOMFLY recursions is justified only by phrases such as "after some simple algebraic manipulations" and "one can easily show". Moreover, the theorem states the result for K_{j,2} and K_{j,3} for all j in Z, but the written induction covers only j >= 0; the negative-j case is dismissed with "the recursive formulas will differ slightly" (p. 21). The proof of part (iii) likewise relies on the unproved link identity (86) and a final step described as "straightforward to compute". If any of the asserted cancellations or the negative-j recursions is incorrect, the conclusion H = dKF does not follow for the full families. These gaps are load-bearing and should be filled.
  3. [Sec. 4.2, Eqs. (97)-(99), Remark 4.2] Equation (99) is not correct as written. From (97), the odd-z part of \hat g is (a-a^{-1})/z gKF_even - gKF_odd - \bar H, which equals (a-a^{-1})/z (dKF - H) because \bar H = (a-a^{-1})/z H and dKF = gKF_even - z/(a-a^{-1}) gKF_odd. The printed expression uses H instead of \bar H and omits the prefactor (a-a^{-1})/z. Similarly, (98) is missing a factor of (a-a^{-1})/z in the identification with ddKF. As a consequence, the claimed equivalence (100) between H = dKF and the vanishing of \hat N^{c=2}_{g,Q} does not follow from the displayed equations. Please correct these formulas and re-verify the subsequent BPS claims, in particular the statement in Remark 4.2.
minor comments (5)
  1. [Throughout] There are several typographical errors, e.g., "Alexader" in Remark 3.1, "factorasibale" in Sec. 2.2, and "invequality" in Remark 2.2; these should be corrected.
  2. [Eq. (97)] The expression "(a-a^{-1}/z - 1)" is ambiguous; please write \frac{a-a^{-1}}{z} - 1 explicitly.
  3. [Theorem 3.1 proof] The proof refers to "the pole of order N at lambda = 0", but the integrand lambda^{-N-1}Z has a pole of order N+1; this is a wording issue that should be clarified.
  4. [Remark 5.3] The statement that, in factorised cases, the Jones polynomial "essentially contains the same information as Z(K)" is too strong, since cancellations can occur between the exponents (as the authors themselves note for j=3 in the table for 5_2); the HZ exponents are not always recoverable from \bar J.
  5. [Sec. 4.1, Conjecture 4.1] The claimed computational verification of the "if and only if" statement for all knots up to 12 crossings is not documented in the paper; providing the list or the computation as supplementary data would make the empirical basis of Conjecture 4.1 checkable.

Circularity Check

1 steps flagged · score 3.0 of 10

Core Theorem 4.1 is independent of the HZ extrapolation; the only definitional reduction is the BPS remark, where Eq. (99) defines the c=2 invariant as dKF-H, making the advertised implication tautological.

  1. self definitional [Sec. 4.2, Eqs. (99)-(100), Remark 4.2]
    "X_{g,Q} \hat N^{c=2}_{g,Q} z^{2g-1}a^Q = \frac{a-a^{-1}}{z} gKF_{\rm even} - gKF_{\rm odd} - H = dKF - H. (99) ... Remark 4.2. According to (99), the relation (62) between the Kauffman and HOMFLY–PT polynomial of a knot is equivalent to the vanishing of the two-crosscaps BPS invariants \hat N^{c=2}_{g,Q}, i.e. H = dKF \Leftrightarrow \hat N^{c=2}_{g,Q} = 0. (100)"

    The c=2 BPS generating function is not computed independently in this paper. Equation (99), together with the proposed parity split of \hat g in (97), defines \sum \hat N^{c=2} z^{2g-1}a^Q to be exactly dKF - H. Remark 4.2 then states that H = dKF is equivalent to the vanishing of these invariants. This is the tautology 'H = dKF iff dKF - H = 0'; the advertised 'peculiar implication in topological string theory' reduces by construction to the paper's own defining formula. The agreement checks against [35] support the proposed identification of \hat g, but they do not give the equivalence (100) independent content.

full rationale

The central mathematical result, Theorem 4.1, is not circular: it proves H = dKF for the listed families by induction from Kauffman skein recursions and previously derived HOMFLY recursions, and it does not use the conjectural HZ-factorisation formulas as an input. The infinite-family HZ formulas (9)-(25) are admittedly extrapolated from finitely many members (footnote 3), and the Kauffman recursions (63)-(72) are stated without derivation; these are explicit rigour gaps rather than circular reductions. The use of [14] for the HOMFLY recursions is a normal load-bearing citation to a published independent result, not a circularity, since the cited recursions are not the target equality. The one genuine definitional reduction is the BPS remark: Eq. (99) defines \hat N^{c=2} to be proportional to dKF - H, so the equivalence in (100) is immediate from the definition. This affects a secondary interpretive claim, not the proof of H = dKF, hence the low score of 3.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper's central claims rest on standard knot polynomial theory plus two non-standard extrapolations: the pattern-based HZ formulas for arbitrary twist powers, and the proposed BPS identification. The HOMFLY-Kauffman proof uses recursion formulas stated without full derivation.

free parameters (1)
  • HZ exponents alpha_i(k), beta_i(k), m for generated families = e.g., 5_2^{3k}: alpha=(12k+3, 12k+13), e=6k+1; see (9)-(25)
    Inferred by extrapolating from the first few members of each family (footnote 3); not derived for all k. These integers are not fitted in a statistical sense, but they are chosen by hand/pattern for arbitrary k.
assumptions (5)
  • domain assumption Factorisability of the HZ transform is preserved under arbitrary powers of full twists F_m and Jucys-Murphy braids E_m.
    Used throughout Sec. 2 to generate infinite families; supported only by finite checks and pattern extrapolation.
  • domain assumption HZ formulas (9)-(25) for the k-th members of families hold for all k.
    Stated in footnote 3 as extrapolation; a follow-up paper is promised.
  • ad hoc to paper The unoriented BPS generating function equals (a-a^{-1}/z - 1)(gKF_even+gKF_odd) - \bar H, split as in (97)-(99).
    Proposed in Sec. 4.2 and checked against [35] for some knots; not derived from LMOV.
  • domain assumption HZ exponents coincide with rows of Khovanov homology tables for factorised knots and links.
    Observed in examples in Sec. 5; only Prop. 5.1 (Jones via residue) is proven.
  • standard math Skein relations for HOMFLY-PT (1) and Kauffman (58), and the geometric series evaluation (4), are valid.
    Standard background for the HZ transform computation.

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Pith. "Pith review of The HOMFLY-PT polynomial and HZ factorisation." pith.science (2026). https://pith.science/paper/6HTMQM2K

@misc{pith2026241204933,
  author       = {Pith},
  title        = {Pith review of: The HOMFLY-PT polynomial and HZ factorisation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6HTMQM2K}},
  note         = {Machine review of arXiv:2412.04933}
}
read the original abstract

The Harer-Zagier (HZ) transform maps the HOMFLY-PT polynomial into a rational function. For some special knots and links, the latter admits a simple factorised form, which is referred to as HZ factorisation. This property is preserved under full twists and the Jucys-Murphy twists, which are hence used to generate infinite HZ-factorisable families of hyperbolic knots. For such families, the HOMFLY-PT polynomial can be fully encoded in two sets of integers, corresponding to the numerator and denominator exponents, which turn out to be related to the double-grading in Khovanov homology. Moreover, a relation between the HOMFLY-PT and Kauffman polynomials, which was only known to hold for torus knots, is now proven for several of these hyperbolic families. Such a relation has a peculiar implication in topological string theory, namely, it is equivalent to the vanishing of the two-crosscap BPS invariants. It is conjectured that the HOMFLY-PT/Kauffman relation provides a criterion for HZ factorisability.

Figures

Figures reproduced from arXiv: 2412.04933 by the authors.

Figure 1
Figure 1. Two operations that preserve the factorisability of the HZ transform. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Relations between HZ-factorisable knots with braid index (a) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. The family 53k 2 is the closure of the braid obtained by a braid repre￾sentative for 5+ 2 (left part) concatenated with k full twists F3 (one of such twists is depicted in the right part). 10+ 139 = 5+ 2 ⊗ F3. In turn, as suggested in the table (a) of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The twisted torus knot T(4, 5, 3, 3k) is the closure of this braid. transform for odd j is Z(T(4, 2j+1, 3, 3k)) = λ [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Hopf link T(2, 2) with parallel relative orientation on its components. Z(T(2, 2j); λ, q) = λ [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

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Reviewed August 11, 2026 · model on record in the stance chip above.