{"id":"eacabbec-a5a9-42bb-9211-8a66de736956","arxiv_id":"2412.04933","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The HOMFLY-PT and Kauffman polynomials satisfy a previously torus-only relation for several infinite hyperbolic knot families that have factorised Harer-Zagier transforms.","lead":"This paper shows that for several infinite families of hyperbolic knots, a known relation between the HOMFLY-PT and Kauffman polynomials, previously only proven for torus knots, also holds. The authors connect this relation to the vanishing of two-crosscap BPS invariants in topological string theory and to a factorisation property of the Harer-Zagier transform.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.1's induction rests on unproved Kauffman recursion formulas, and the extension to negative j is only asserted; the key cancellations are not demonstrated.","rationale":"The reader correctly identified the extrapolated HZ formulas and the unproved Kauffman recursions as the main weaknesses. I focus on the latter because Theorem 4.1's proof is an algebraic induction that depends entirely on the Kauffman recursions and the asserted cancellations; the HZ factorised formulas are not used in the proof of the HOMFLY-Kauffman relation itself. The paper does contain genuine independent support: Theorems 3.1 and 3.2 are proven with contour integrals, and the finite checks across many knots and links are extensive and reproducible. However, the induction in Theorem 4.1 is incomplete as written: the recursive formulas are asserted, the cancellations are not shown, and the negative-j cases are only promised. This does not mean the theorem is false — it may well be true — but the proof, as presented, leaves a specific, checkable gap. A symbolic verification for negative j (and for a few positive j) would settle whether the gap is merely cosmetic or hides a real error. The reader's CONDITIONAL verdict is appropriate; my analysis does not change it.","tokens_in":41636,"tokens_out":6448,"duration_ms":69479,"concrete_test":"Use a computer algebra system (e.g. Mathematica with the KnotTheory` package) to compute the Dubrovnik Kauffman polynomial of K_{-1,2} and K_{-2,2} directly from the skein relation, and symbolically verify whether the asserted recursion (68) and the cancellation leading to Eq. (69) hold for these negative j values. A nonzero residual term would falsify Theorem 4.1(ii) for the advertised range j ∈ Z.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 4.1 is an induction that depends on recursive formulas for the Dubrovnik Kauffman polynomial: (63), (65), (68), (70), (72). These are stated without derivation from the skein relation (58), and the reduction to the HOMFLY recursions (64), (66), (69), (71), (73) is justified only by phrases such as 'after some simple algebraic manipulations' and 'one can easily show that ... exactly cancel'. In particular, for the families K_{j,2} and K_{j,3}, the theorem is stated for all j ∈ Z, but the written induction covers only j ≥ 0; the case j < 0 is dismissed with 'the recursive formulas will differ slightly' (p. 21). If any of these asserted cancellations contains a sign or coefficient error, or if the negative-j recursion does not preserve the cancellation, the conclusion H = dKF is not established for the infinite family. This is the most load-bearing gap for the central claim: while the HZ extrapolation is admitted in footnote 3, Theorem 4.1's proof does not actually use the factorised HZ formulas, so the unproved Kauffman recursions are the true engine of the theorem and the least secure part of the argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":41967,"tokens_out":27079,"duration_ms":223032,"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":[{"comment":"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.","section":"Sec. 2, footnote 3, Eqs. (9)-(25)"},{"comment":"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.","section":"Sec. 4.1, Eqs. (63)-(73), proof of Theorem 4.1"},{"comment":"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.","section":"Sec. 4.2, Eqs. (97)-(99), Remark 4.2"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"The expression \"(a-a^{-1}/z - 1)\" is ambiguous; please write \\frac{a-a^{-1}}{z} - 1 explicitly.","section":"Eq. (97)"},{"comment":"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.","section":"Theorem 3.1 proof"},{"comment":"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.","section":"Remark 5.3"},{"comment":"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.","section":"Sec. 4.1, Conjecture 4.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript mixes rigorous statements with extensive conjectural and computational material. The presentation often labels extrapolated formulas as established facts, which needs correction. The core ideas are interesting and likely of value to the mathematical physics community, but the paper would benefit from a clearer separation between theorems and conjectures, and from completing or honestly downgrading the unproved parts of Theorem 4.1."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe main new thing here is the systematic extension of HZ factorisation from the authors' earlier single pretzel family to several new infinite hyperbolic families, using full twists and Jucys–Murphy braids. The contour-inversion results (Theorems 3.1 and 3.2) are clean and genuinely useful, and the observation that HZ exponents coincide with Khovanov row gradings (Prop. 5.1) is a neat structural insight. The authors are also honest about the limits: footnote 3 explicitly says the infinite-family HZ formulas are extrapolated from small members, and a follow-up paper is promised.\n\nThe soft spots are real but manageable. The proof of Theorem 4.1 rests on recursive formulas for the Kauffman polynomial that are stated without derivation; phrases like “after some simple algebraic manipulations” hide substantial cancellations, and the extension to negative j is only asserted. If any of those cancellations has a sign or coefficient error, the induction collapses. As written, this part is a proof sketch rather than a complete proof. The HZ extrapolation is a separate issue, and it is worth noting that Theorem 4.1 does not actually use the factorised HZ formulas, so a failure of the extrapolation would not harm the Kauffman relation for the recursively proven families, though it would remove the claimed application to those families.\n\nThe BPS interpretation is more interpretive than predictive. The proposed identification (97)–(99) makes H = dKF equivalent to vanishing c = 2 invariants by construction, so the “peculiar implication” is not an independent physical prediction. The authors do match existing data, which makes it a reasonable suggestion, but the reader should not mistake it for a derivation.\n\nWho is this for? People working on knot polynomials, HZ-type transforms, and Khovanov homology; topological string people may find the BPS side suggestive, but it is not the core. The paper deserves a serious referee. The deficiencies are of detail and exposition rather than conception, and the authors have documented the gaps themselves. I would recommend acceptance after revision, with the referee asking for the full Kauffman recursion derivations, or at least the omitted algebra in an appendix, and a precise treatment of the j < 0 cases.","headline":"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.","tokens_in":42459,"tokens_out":2515,"would_cite":true,"duration_ms":28596,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K10","57K14","57K18"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["HOMFLY-PT polynomial","Harer-Zagier transform","HZ factorisation","Kauffman polynomial","hyperbolic knots","Jucys-Murphy braid","Khovanov homology","BPS invariants"],"falsifier":"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.","tokens_in":41450,"feed_emoji":"🔗","tokens_out":9972,"duration_ms":88965,"temperature":0.7,"pith_summary":"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.","feed_headline":"One identity pins the HOMFLY-PT polynomial to the Kauffman polynomial","feed_subtitle":"For four infinite hyperbolic families, it also forces the two-crosscap BPS invariants to vanish.","key_machinery":"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 $-\\sum_i q^{\\alpha_i}+\\sum_j q^{\\beta_j}$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces the Harer–Zagier transform; the paper's central object derives from this generating-function construction.","marker":"[5]"},{"why":"Establishes factorised HZ transforms for torus knots and connects the HZ transform to knot matrix models; the starting point extended here.","marker":"[13]"},{"why":"The authors' earlier paper found the first hyperbolic pretzel family with factorised HZ transform and gave the recursion used in Theorem 4.1.","marker":"[14]"},{"why":"Proves the HOMFLY–PT/Kauffman relation for torus knots; the statement that Theorem 4.1 generalises to hyperbolic families.","marker":"[15]"},{"why":"Supplies the knot invariant data used for the exhaustive up-to-12-crossing checks behind Conjecture 4.1.","marker":"[23]"},{"why":"Provides the BPS invariant computations and tables whose two-crosscap values are compared with the vanishing predicted by (99).","marker":"[35]"},{"why":"Supplies the Khovanov homology tables whose row exponents the factorised HZ exponents are matched against in Section 5.","marker":"[17]"},{"why":"Defines Khovanov homology, the categorification whose graded Euler characteristic appears in Proposition 5.1.","marker":"[18]"}],"fun_headline_variants":["HZ factorisation extends to hyperbolic knots, linking two polynomials","HOMFLY-PT equals Kauffman for HZ-factorisable hyperbolic knots","Two-crosscap BPS invariants vanish for HZ-factorisable knots","HZ factorisation ties HOMFLY-PT and Kauffman for hyperbolic knots","Hyperbolic twists generate HZ-factorisable knots with vanishing BPS invariants"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["HZ factorisation extends to hyperbolic knots, linking two polynomials","HOMFLY-PT equals Kauffman for HZ-factorisable hyperbolic knots","Two-crosscap BPS invariants vanish for HZ-factorisable knots","HZ factorisation ties HOMFLY-PT and Kauffman for hyperbolic knots","Hyperbolic twists generate HZ-factorisable knots with vanishing BPS invariants"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00148,"raw_usage":{"total_tokens":6028,"prompt_tokens":1111,"completion_tokens":4917,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":727,"completion_tokens_details":{"reasoning_tokens":4815}},"tokens_in":727,"tokens_out":4917,"duration_ms":31575,"temperature":1.0,"reasoning_tokens":4815,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:07:18.677756+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Morozov, A","cited_arxiv_id":null,"evidence_quote":"Establishes factorised HZ transforms for torus knots and connects the HZ transform to knot matrix models; the starting point extended here."},{"cited_title":"Petrou and S","cited_arxiv_id":null,"evidence_quote":"The authors' earlier paper found the first hyperbolic pretzel family with factorised HZ transform and gave the recursion used in Theorem 4.1."},{"cited_title":"Livingston and A","cited_arxiv_id":null,"evidence_quote":"Supplies the knot invariant data used for the exhaustive up-to-12-crossing checks behind Conjecture 4.1."},{"cited_title":"Mironov, A","cited_arxiv_id":null,"evidence_quote":"Provides the BPS invariant computations and tables whose two-crosscap values are compared with the vanishing predicted by (99)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Khovanov homology tables whose row exponents the factorised HZ exponents are matched against in Section 5."},{"cited_title":"Khovanov, A categorification of Jones polynomial, Duke Math J","cited_arxiv_id":null,"evidence_quote":"Defines Khovanov homology, the categorification whose graded Euler characteristic appears in Proposition 5.1."}],"review_version":1}