{"id":"f417fa6f-6943-4b57-b83a-c3092ec4e3bf","arxiv_id":"2607.21479","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The SU(N) homological block in inverted-Habiro form equals a half-index of an explicit 3d N=2 theory for the figure-eight and the two trefoil knots, with a conjectural all-knot extension.","lead":"For three example knots, this paper builds explicit 3d supersymmetric field theories whose half-index partition function reproduces the SU(N) homological block of the knot complement, and gives a recipe to read SU(N) homological blocks from colored HOMFLY-PT polynomials. The extension to all knots is stated as an expectation; the worked examples are cross-checked against published results.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central claim rests on unproven conjecture (2.3); examples have trivial Habiro coefficients, so the half-index may not realize the true homological block for knots with nontrivial α.","rationale":"The reader identifies the same weakest assumption: the unproven conjecture (2.3) that the SU(N) homological block is the inverted-Habiro series. This is indeed the single most load-bearing premise, because every half-index realization in the paper is a realization of that series. The three examples all have trivially simple inverted Habiro coefficients, so they provide no nontrivial evidence for the conjecture; they only check internal consistency. Thus the central claim is conditional on an external result, exactly as the reader's CONDITIONAL verdict reflects. I considered alternative concerns (incomplete superpotential in footnote 3, contour ambiguity for the Weyl partner, non-uniqueness of Θ_z), but these are either explicitly acknowledged or do not affect the three worked examples. The (2.3) conjecture is more fundamental: if false, the paper's main construction would not be about homological blocks at all. The proposed test—checking the conjecture on a knot with nontrivial Habiro coefficients—would directly address this concern. Since the reader's verdict already captures this conditionality, no change is needed.","tokens_in":37282,"tokens_out":10688,"duration_ms":99518,"concrete_test":"Select a knot with known nontrivial Habiro coefficients, e.g., 5_2 or 6_1 (data in [18]). Compute the inverted-Habiro coefficients α_{-k-1} from the quantum C-polynomial (or, if available, from [14]), form the series (2.4), and test: (i) annihilation by the known quantum A-polynomial from [18]; (ii) agreement of the x-expansion with the SU(2) and SU(3) homological block expansions in [18] to order q^10. If either fails, the conjecture (2.3) is false for that knot and the half-index realization does not give the homological block. If it passes for several nontrivial knots, the conditional is strengthened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central assertion is that the SU(N) homological block for knot complements is realized as a half-index. The entire construction is built on the conjectural template (2.3) from [14]: that the homological block equals the inverted-Habiro series with coefficients α_{-k-1} and q-Pochhammer factors (x;q)_{k+N-1}(x^{-1}q^{2-N};q)_{k+N-1}. This conjecture is not proven in the paper, and the three examples do not test it: the figure-eight has α=1, and the trefoils have monomial α (2.39), (2.60). These are the simplest possible inverted Habiro coefficients, so agreement with [18]'s expansions does not discriminate between the conjecture and other series with the same leading asymptotics. If (2.3) fails for a knot with nontrivial α (e.g., 5_2 or 6_1), then the half-index integrals (2.24), (2.41), (2.62) would produce a series that is not the knot's homological block, invalidating the advertised 3d-3d realization for that knot. Additionally, the Θ_z interpolation of §2.2, the proposed route to general knots, requires constructing a function from data on the lattice z=q^{1-N-k} to values at z=q^k; the paper gives examples but no existence or uniqueness argument, so the method is an ansatz rather than a proven algorithm. These concerns do not undermine the internal consistency of the three worked examples, but they mean the central claim is conditional on an external, untested premise.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends the author's earlier SU(2) half-index realization of knot-complement homological blocks to SU(N) for the totally symmetric representation. For the figure-eight and the left/right trefoil knots, it constructs explicit contour integrals whose residues at one family of poles reproduce the inverted-Habiro form of the SU(N) homological block (Eqs. (2.5), (2.40), (2.61) with a=q^N) and whose residues at another family reproduce the colored HOMFLY-PT polynomial (Eqs. (2.28), (2.37)). The integrands are interpreted as half-indices of 3d N=2 theories with stated field content and superpotential (Eq. (2.27) and surrounding text). Section 2.2 proposes a general method, via an interpolation factor Θ_z, to obtain the inverted Habiro coefficients from Habiro-series data of the HOMFLY-PT polynomial. The paper also computes a-deformed versions, quantum A/B-polynomial annihilators, partition functions on S^2×_q S^1 and S^3_b, twisted indices on M_{g,p}, and proposes GL(L|M) homological blocks by specializing a=q^{L-M}.","tokens_in":37654,"tokens_out":10455,"duration_ms":90735,"significance":"If the central conjecture (2.3) holds, the paper provides the first SU(N) half-index realization of knot-complement homological blocks in inverted-Habiro form, together with a practical route from HOMFLY-PT Habiro data to the homological block. The strengths are substantial: the worked examples are explicit and internally consistent; the residue computations are detailed; the resulting quantum A- and B-polynomials are matched against externally published results ([18], [21]); there are no fitted parameters; and the M-theory/large-N interpretation adds a useful conceptual frame. The paper is honest about its limitations, including footnote 3 on the incomplete UV superpotential and footnote 6 on the interpolation factor. Nevertheless, the advertised extension to general knots rests on an unproven conjecture and on examples whose inverted Habiro coefficients are trivial or monomial, so the generality claim is currently unsupported.","major_comments":[{"comment":"The entire construction is built on the conjectural template (2.3) from [14], namely that the SU(N) homological block equals the inverted-Habiro series with coefficients α_{-k-1} and Pochhammer factors (x;q)_{k+N-1}(x^{-1}q^{2-N};q)_{k+N-1}. The paper does not prove this conjecture, and its three examples do not test it: the figure-eight has α=1 and the trefoils have monomial α (2.39), (2.60). Agreement with [18]'s positive expansions for these cases cannot distinguish the conjecture from other series with the same leading asymptotics. If (2.3) fails for a knot with nontrivial α, the half-index integrals (2.24), (2.41), (2.62) would realize a series that is not the knot's homological block. This is a correctness risk, not an internal inconsistency, but it is load-bearing for the paper's main claim to provide a general 3d-3d realization. A concrete test using a knot with nontrivial Habiro","section":"§2.1, Eq. (2.3)"},{"comment":"The proposed method for obtaining the homological block from a HOMFLY-PT Habiro series relies on the existence of the interpolation factor Θ_z. Equation (2.38) fixes Θ_z only on the pole family z=q^{1-N-k}; evaluating it at the other family z=q^k requires an interpolation across the z-plane. The paper gives no existence or uniqueness argument, and footnote 6 concedes that Θ_z may need to be expressed as a contour integral or may contain q-Pochhammer factors. Thus the method is an ansatz, not a proven algorithm. This directly affects the abstract's promise of 'a method for obtaining the G=SU(N) homological block and its a-deformed version ... from a Habiro series expression.' The authors should either prove a canonical construction for a nontrivial example or sharply delimit the claim to the cases where Θ_z is explicitly exhibited.","section":"§2.2, Eq. (2.38) and footnote 6"},{"comment":"The 3d N=2 theory T[M_3] is not fully specified. The superpotential (2.27) preserves all global symmetries whose fugacities appear in the half-index, but the paper acknowledges in footnote 3 that the additional couplings needed to break extraneous global symmetries are not included, and the corresponding fugacities are simply turned off by hand. The claim that the half-index 'realizes' the homological block as a half-index of T[M_3] is therefore conditional on the existence of a UV completion whose protected quantities coincide with those computed here. This is a self-acknowledged gap; it should be addressed or at least stated as a standing assumption in the main text, not only in a footnote, because it affects the uniqueness and physical interpretation of the engineered theories.","section":"§2.1, footnote 3 and Eq. (2.27)"}],"minor_comments":[{"comment":"Please check the placement of the Pochhammer ratios: (q^{k+N-1}x;q)_∞/(x;q)_∞ equals 1/(x;q)_{k+N-1}, which is the inverse of the factor (x;q)_{k+N-1} appearing in the template (2.4). If this is intentional (i.e., the homological block has the inverse Pochhammers), the relation to (2.4) should be explained; otherwise it is a typo.","section":"Eq. (2.5)"},{"comment":"The symbol ≃ is used to denote equality up to theta-function ambiguity (footnotes 7 and 12). This is a crucial caveat; please define it once in the main text before first use and state explicitly how the ambiguity affects the claimed 'realization'.","section":"§2.3, Eqs. (2.42), (2.63)"},{"comment":"The anti-half-index/mirror relation (3.20) is checked only 'up to theta function ambiguity and q-dependent factors.' For a paper that emphasizes exact correspondences, this should be stated as a conjecture or the precise factors should be tabulated for the three examples.","section":"§3.1, Eq. (3.20)"},{"comment":"There are several minor typographical issues: 'the the totally symmetric representation' in §2; equation numbering for the alternative a-deformed forms (2.43) and (2.64) could be referenced more clearly; and footnote 2's caveat that the Gröbner basis may generate only a subideal of the dAB-ideal should be repeated where those ideals are quoted as results.","section":"§2; general"},{"comment":"Reference [15] is cited as '2603.05236' with a future date; please ensure the published/updated version is cited. Also, the usage of [18] for positive expansions and [21] for quantum A-polynomials is appropriate and should be kept.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is a careful and honest execution of a conjectural program, and the worked examples are internally solid. The main risk is the gap between the examples—which have trivial or monomial inverted Habiro coefficients—and the advertised general method. I would not reject, because the internal consistency and the external checks against [18] and [21] justify publication after revision. However, the authors should either add a nontrivial knot test (e.g., 5_2 or 6_1) or explicitly state in the abstract and conclusions that the realization is established only for the listed cases and that the general claim is conjectural. The incomplete UV superpotential and the non-uniqueness of Θ_z should also be promoted from footnotes to visible caveats. If the authors are willing to make those changes, the paper would be a solid contribution to the 3d-3d literature."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is a careful, honest paper. It extends the SU(2) half-index construction to SU(N) by packaging the N-dependence as a=q^N, and gives explicit a-deformed half-index integrands for the figure-eight and both trefoils. The algebra checks out: the residues reproduce the homological blocks and the colored HOMFLY-PT polynomials, and the quantum A- and B-polynomials agree with [18] and [21]. No fitted parameters anywhere. The caveats—that (2.3) is a conjecture, that the UV superpotential is incomplete, that the Θ_z interpolation is an ansatz—are stated in footnotes rather than hidden. The partition-function analysis at a=q^N is a genuine extra, not just an application.\n\nSecond, the stress-test concern is real and should be in the referee report. The whole construction is built on the inverted-Habiro template (2.3) from Park's conjecture. The three examples have the simplest possible Habiro coefficients: α=1 for the figure-eight, monomials for the trefoils. Agreement with [18]'s positive expansions does not discriminate between the conjecture and other series with the same leading asymptotics, because the coefficients aren't being exercised. If (2.3) fails for a knot like 5_2 or 6_1, the half-index integrals would produce a series that is not the homological block. The paper does not close that gap, and the Θ_z recipe of Section 2.2 is an existence ansatz rather than a proven algorithm. None of this undermines the internal consistency of the three worked examples, but it does mean the advertised '3d-3d realization for every knot' is conditional on an external premise.\n\nMinor soft spots: the computer-algebra results are reported without code or notebooks, which slows spot-checking. The GL(L|M) proposal is a one-line specialization and is not tested. The missing symmetry-breaking couplings (footnote 3) mean the '3d N=2 theory' is defined only through its protected quantities.\n\nBottom line: this deserves a serious referee. It's a sound construction for simple knots, honestly limited, and the conjectural template is from the literature. A referee should ask for a nontrivial example with non-trivial Habiro coefficients, or an argument that the identification (2.38) is canonical. I'd cite the figure-eight a-deformed block and bring it to reading group.","headline":"Solid, honest construction of SU(N) homological blocks as half-indices for three knots, but the central identification is conditional on a conjecture the examples don't actually test.","tokens_in":38278,"tokens_out":2401,"would_cite":true,"duration_ms":22148,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that the SU(N) homological block for a knot complement, expressed as an inverted Habiro series, is the half-index of a 3d N=2 theory, and that the same contour integral yields the colored HOMFLY-PT polynomial when a differ","keywords":["3d-3d correspondence","homological block","half-index","Habiro series","HOMFLY-PT polynomial","knot complement","SU(N) Chern-Simons","3d N=2 theory"],"falsifier":"Apply the paper's $\\Theta_z$ prescription to a knot not treated in the paper (e.g., the $5_2$ knot): compute the inverted Habiro coefficients from its colored HOMFLY-PT Habiro series, build the half-index integral, and compare the resulting $q$-series numerically at fixed $N$ and $x$ with the independent expansion from the state-integral or the quantum C-polynomial approach; any mismatch at some order $q^m$ would falsify the claimed identification.","tokens_in":37014,"feed_emoji":"🌀","tokens_out":3344,"duration_ms":30903,"temperature":0.7,"texified_at":"2026-08-05T21:39:18.014742+00:00","pith_summary":"The paper aims to show that for gauge group $\\mathrm{SU}(N)$ and the totally symmetric representation, the homological block of a knot complement arises as a half-index of a 3d $N=2$ theory. The same half-index integral, with a different choice of poles, reproduces the colored HOMFLY-PT polynomial, so the Habiro data of the polynomial determines the $\\mathrm{SU}(N)$ homological block. This is worked out for the figure-eight and both trefoil knots, with explicit field content and superpotential, and is expected to extend to general knots. The author also derives $a$-deformed versions, which encode the $N$-dependence via $a=q^N$, and uses them to discuss large-$N$ limits, supergroup extensions, and partition functions.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":6626,"prompt_tokens":801,"completion_tokens":5825,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":801,"completion_tokens_details":{"reasoning_tokens":5030}},"feed_headline":"One half-index yields both knot block and HOMFLY polynomial","feed_subtitle":"Choosing different poles in the same contour integral gives the SU(N) homological block or the colored HOMFLY-PT invariant, for all N.","key_machinery":"The key object is the half-index contour integral $ (q;q)_\\infty \\oint \\frac{dz}{2\\pi i z} \\Upsilon(z,x,a,q) $ with $\\Upsilon$ built from $q$-Pochhammer symbols and theta functions. The choice of poles inside the contour selects which invariant is produced: poles from $(z^{-1};q)_\\infty^{-1}$ give the homological block, while poles from $(q^{-1} a z;q)_\\infty^{-1}$ give the colored HOMFLY-PT polynomial. The interpolation factor $\\Theta_z$, whose value at the $z=q^k$ and $z=q^{1-N-k}$ poles encodes the relation between Habiro and inverted Habiro coefficients, is the mechanism that turns a HOMFLY-PT Habiro series into the $\\mathrm{SU}(N)$ homological block.","core_discovery":"The central claim is that the $G=\\mathrm{SU}(N)$ homological block for a knot complement, given in inverted Habiro form, can be written as the contour integral $ (q;q)_\\infty \\oint \\frac{dz}{2\\pi i z} \\Upsilon_K(z,x,a,q) $ with $a=q^N$, where taking the poles $z=q^k$ ($k\\ge 0$) yields the homological block and taking the poles $z=a^{-1}q^{1-k}$ yields the colored HOMFLY-PT polynomial. For the figure-eight knot, the integrand $\\Upsilon$ is explicitly constructed from $q$-Pochhammer symbols; the 3d $N=2$ theory has seven chiral multiplets with specified charges and a superpotential. The same pattern holds for the left- and right-handed trefoil knots. The author further shows how, given a Habiro series for the HOMFLY-PT polynomial, one can extract the invert","pith_inferences":["The method suggests that the Habiro-series form of the HOMFLY-PT polynomial, which is known for many knots, can be systematically converted into a homological block; testing this on a knot not treated here (e.g., the 5_2 or 6_1 knot) would be a straightforward numerical check of the conjecture.","The theta-function ambiguity in the a-deformed homological blocks may be an artifact of the half-index presentation; the a=q^N specialization, where the ambiguity disappears, likely fixes a canonical normalization.","The observation that the abelian branch contributes zero relative to the non-abelian branch at a=q^N in partition functions suggests that, for open 3-manifolds with non-abelian flat connections, homological blocks alone do not give the full partition function, in contrast to the closed 3-manifold case.","The existence of a half-index for every knot whose HOMFLY-PT polynomial has the conjectured Habiro form would imply that the 3d-3d correspondence for knot complements holds at arbitrary N, not just in the large-N limit."],"forward_implications":["If the construction is correct, any colored HOMFLY-PT polynomial with a totally symmetric representation and a Habiro series determines the SU(N) homological block and its a-deformed version for the corresponding knot complement.","The explicit 3d N=2 theories T[S^3\\K, SU(N)] obtained for these knots capture both abelian and non-abelian flat connections, providing a complete description in the 3d-3d correspondence.","The a=q^N packaging makes the large-N limit manifest, and the geometric transition to a resolved conifold gives a physical interpretation of the a-deformed block as an open topological string partition function.","Partition functions on S^2×_q S^1 and S^3_b factor into half-index and anti-half-index pairs, and the anti-half-index for a knot is related to the half-index of its mirror knot, extending homological blocks from |q|<1 to |q|>1.","The specialization a=q^{L-M} produces homological blocks for supergroup GL(L|M) Chern-Simons theory, extending the correspondence beyond ordinary Lie groups."],"fun_headline_variants":["Half-index encodes homological block and HOMFLY","One contour integral gives two knot invariants","3d theory yields knot complement invariants via poles","From Habiro to half-index: unifying knot invariants","Pole selection in half-index extracts HOMFLY"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The construction rests on the conjecture that the $\\mathrm{SU}(N)$ homological block is given by the inverted Habiro series with the specific $q$-Pochhammer prefactors of equation (2.3); if that identification fails for some knot or some $N$, the half-index derived here is a realization of a different series.","fun_headline_variants_meta":{"raw":{"variants":["Half-index encodes homological block and HOMFLY","One contour integral gives two knot invariants","3d theory yields knot complement invariants via poles","From Habiro to half-index: unifying knot invariants","Pole selection in half-index extracts HOMFLY"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000261,"raw_usage":{"total_tokens":1429,"prompt_tokens":744,"completion_tokens":685,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":488,"completion_tokens_details":{"reasoning_tokens":608}},"tokens_in":488,"tokens_out":685,"duration_ms":6462,"temperature":1.0,"reasoning_tokens":608,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T07:19:51.998888+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Apply the paper's $\\Theta_z$ prescription to a knot not treated in the paper (e.g., the $5_2$ knot): compute the inverted Habiro coefficients from its colored HOMFLY-PT Habiro series, build the half-index integral, and compare the resulting $q$-series numerically at fixed $N$ and $x$ with the independent expansion from the state-integral or the quantum C-polynomial approach; any mismatch at some order $q^m$ would falsify the claimed identification.","supporting_citations":[],"review_version":1}