{"id":"12a4804c-4ad0-491b-9d27-0e15d24df744","arxiv_id":"1908.06085","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A systematic relator based computation produces new integer-valued invariants of spherical curves under Reidemeister-type moves, with explicit examples up to six arrows.","lead":"This paper introduces a computer-assisted method to construct integer-valued invariants of oriented spherical curves, using arrow diagrams and algebraic relators. It yields the first explicit invariants under weak (1,3) homotopy for diagrams with up to six arrows.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Computational section solves a different linear system than Theorem 1: matrix uses x*_i (subword counts) rather than x̃*_i, so reported invariants and dimensions are not justified by the theorem.","rationale":"The reader's concern was enumeration completeness. Although that is relevant, the more fundamental issue is that the matrix in the computational section appears to evaluate the descendant functions x*_i (subword counts) on the relators, whereas Theorem 1 requires the primitive functions x̃*_i on the relators. In the proof of Theorem 1, the difference of the invariant under a move is expressed as a sum of x̃*_i evaluated on relators built from each sub-context; the condition needed is L(r)=0 for all relators r, not L(E(r))=0. The entry 3 in Example 9's first row confirms the computation uses x*, because for a strong-RIII relator in range (2,3) the primitive evaluation takes values in {-1,0,1}. Since the two linear systems are related by an invertible change of basis, the dimensions of the kernels coincide, so the dimension claims (e.g., 13 and 31) may survive, but the specific coefficient tuples are not the ones Theorem 1 yields, and the paper does not prove that the displayed functions are invariants. This undermines the computational main results and the tables, independent of whether the enumeration is complete. The paper can be repaired by recomputing with x̃ or by proving Span(E(R)) ⊆ Span(R); until then, the computational claims are conditional.","tokens_in":54974,"tokens_out":37024,"duration_ms":327238,"concrete_test":"Recompute the kernels for Examples 9 and 10 using the corrected matrix B_{ij}=x̃*_i(r*_sj) for the listed relators, and compare the solution space with the reported tuples (the three vectors in (7) and the weak-(1,3) vector (-1,1,-1,-1,1,-1,3)). If these vectors are not in the kernel of B, the computational section does not implement Theorem 1. A secondary check: verify the reported strong-RIII invariants directly on the explicit two-curve sequence in Fig. 16; equality should hold if they are invariant.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 1's hypothesis is that Σ α_i x̃*_i vanishes on the relators (Section 2.3). The computation in Examples 9–10 instead forms A=(x*_i(r*_sj)) and solves α^T A=0. But x*_i(r)=x̃*_i(E(r)) where E(r) is the formal sum of all subwords of the diagrams in r. Since E(r) is not a relator, α^T A=0 is not equivalent to the theorem's condition. This is not a notational slip: in Example 9, the first row of A is (3,0,0,0,0). The relator r*_s1 in R00010(2,3) has S,T,U empty, so it contains exactly two terms of size 3 (one with positive sign, one with negative), forcing x̃*_1(r*_s1) ∈ {-1,0,1}; the entry 3 is a subword count. Thus the reported coefficient tuples and dimensions (including dim V_ori_w(5,6)=31) solve a different system and are not established by Theorem 1.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces integer-valued functions on oriented spherical curves of the form Σ α_i x*_i, where x*_i(AD) counts the number of sub-arrow-diagrams of AD isomorphic to a fixed arrow diagram x*_i. It also defines auxiliary functions Σ α_i x̃*_i that read off coefficients in the free Z-module generated by arrow diagrams, and it defines five families of relators corresponding to the deformations RI, strong RII, weak RII, strong RIII, and weak RIII. Theorem 1 states that if Σ α_i x̃*_i vanishes on all projected relators of the selected deformation types, then Σ α_i x*_i is an invariant under those deformations. The proof is carried out explicitly for RI, strong RII, and strong RIII; the weak RII and weak RIII cases are omitted as \"essentially the same.\" The computational part of the paper enumerates normal oriented Gauss words, reports kernel dimensions of a matrix A=(x*_i(r*_j)), and in particular claims dim V_ori_w(5,6)=31 and an explicit weak (1,3) homotopy invariant; the raw data for n=5,6 is posted on an external webpage.","tokens_in":55219,"tokens_out":25087,"duration_ms":230304,"significance":"If the results stand, the paper provides a clean and systematic machine-assisted source of sub-arrow-diagram count invariants for spherical curves, including invariants under weak (1,3) homotopy, and it gives concrete computational evidence relevant to the Kamada–Nakanishi Question 1. The theoretical criterion in Theorem 1 is attractive, and the written proofs for the RI, strong RII, and strong RIII cases are direct combinatorial cancellations that are easy to follow. The paper also deserves credit for printing explicit matrices, relators, and invariants for small arrow numbers rather than leaving everything to opaque code. However, the central computational claim currently rests on a linear system that does not match the hypothesis of Theorem 1, and it is not accompanied by audited code or by complete data for the largest computation, so the reported dimensions and invariants are not yet established.","major_comments":[{"comment":"The matrix A is defined as A=(x*_i(r*_sj)), and the reported invariants are obtained from α^T A=0. But Theorem 1 requires Σ α_i x̃*_i(r*)=0 for every relator r*, where x̃*_i(r*) is the coefficient of the diagram x*_i in the linear combination r*. The quantity x*_i(r*) is different: it is the signed sum, over all terms of r*, of the number of sub-diagrams of each term isomorphic to x*_i. These two functionals are not equal. For example, in the strong-RIII relator of Definition 7 with S=T=U=∅, the 2-arrow diagram [[¯ij i¯j]] has coefficient +1 in the relator, but it also occurs as a sub-diagram of the 3-arrow term [[¯ij¯ki¯jk]], so x*([[¯ij i¯j]])(r)=2 while x̃*([[¯ij i¯j]])(r)=1. Thus the matrix solved in Examples 9 and 10 is not the coefficient matrix required by Theorem 1 unless an additional transformation is supplied, and no such transformation is mentioned. The reported dimensions, including dim V_ori_w(5,6)=31, and the displayed weak (1,3) homotopy invariant are therefore not justified as stated by the paper's own theorem.","section":"Section 4.2, Examples 9–10"},{"comment":"The proofs for weak RII and weak RIII are omitted with the remark that the arguments are essentially the same as the strong cases. Since weak RII and weak RIII are part of the statement of Theorem 1 and are central to the paper's stated goal of obtaining weak (1,3) homotopy invariants, this is a gap in the proof of a load-bearing result. The authors should either include the omitted arguments or state and prove a precise reduction showing that the weak cases follow word-for-word from the strong cases.","section":"Section 3, cases ǫ3=1 and ǫ5=1"},{"comment":"The completeness and duplicate-freeness of the enumeration of normal oriented Gauss words in Steps 1–5 is asserted without proof or machine-checkable certificate. The claims dim V_ori_w(4,5)=13 and dim V_ori_w(5,6)=31 depend entirely on this enumeration being exact, and the raw list for n=5,6 is only available at an external webpage rather than in the paper or in an audited program. Even after the matrix question in Examples 9–10 is resolved, the computational results need a reproducibility mechanism, such as the source code, the raw enumeration, or a certificate that can be checked independently.","section":"Section 4.1–4.2 and Section 5"}],"minor_comments":[{"comment":"The abstract says that vanishing of Σ α_i x̃*_i implies Σ α_i x̃* is invariant, but the invariant is Σ α_i x*_i; this is presumably a typo and should be corrected.","section":"Abstract and Theorem 1"},{"comment":"The definition of a sub-word appears to define only suffixes of prefixes of w, namely u(j)=w(n-p+j), whereas Definition 3 and the rest of the paper use sub-words obtained by deleting pairs of letters. Please correct the formal definition to match the intended deletion operation.","section":"Definition 1"},{"comment":"The text says the computation uses C++11 and Mathematica, but no code, version numbers, or numerical linear algebra details are provided; adding this information would substantially improve reproducibility.","section":"Section 4.1"},{"comment":"The thirteen invariant tuples in Tables 7–19 are printed without a clear statement of which column index corresponds to which diagram in Tables 4–6; a short example explaining how to read the tuples would make the tables usable.","section":"Tables 7–19"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a relevant question in knot projection theory and has a sound theoretical skeleton, but the computational section currently solves a different linear system from the one in Theorem 1, and the weak RII/weak RIII proofs are omitted. I recommend asking the authors to recompute the matrix with x̃*_i coefficients (or prove an invertible triangular relation between the two matrices), to provide the missing proofs, and to make the enumeration auditable. If the reported invariants survive recomputation, the paper would be a useful contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nTwo things to know about arXiv:1908.06085. First, the relator framework is a genuinely useful way to generate invariants of spherical curves under any of the 32 equivalence relations from the five move types. Theorem 1 is a clean sufficient condition: vanish on the relators with the tilde functions and you get an invariant. The proof for RI, strong RII, and strong RIII is direct and looks correct; the weak RII and weak RIII cases are omitted as “essentially the same,” which is a minor but real gap.\n\nSecond, the computational section is not as cleanly tied to Theorem 1 as it should be. The stress-test note that reached me claims Example 9 solves a different system because the matrix A uses x*_i (subword counts) instead of x̃*_i. I checked the example: that particular claim is wrong. In R00010(2,3) the relators are projected to diagrams with at most three arrows, and the x_i are the connected three-arrow diagrams, so for these terms x_i(r) is just the coefficient of x_i in r—same as x̃*_i(r). The entry 3 in the first row is a multiplicity, not a subword count, and the relator does not have empty S,T,U as the note says.\n\nBut the general issue survives. For b≥3, ČG_{b,b+1} includes diagrams of two different sizes, and a relator term with b+1 arrows can contain several sub-diagrams of size b. Then x_i(r) genuinely differs from the coefficient of x_i in r, and the kernel of the matrix is not obviously the same as the kernel of the tilde evaluation used in Theorem 1. The paper doesn’t address this. It may be that the code actually computes coefficients and the dimension table is fine, but the written definition of the matrix and the appeal to Corollary 2 need explicit justification.\n\nThe other soft spots are minor: the enumeration in Section 4 needs a proof of completeness, and the data for n=5,6 lives on an external webpage with no shipped code. For a theorem-driven computation paper, that’s a reproducibility issue, not a fatal one.\n\nWho it’s for: people working on knot projections or finite-type invariants from Gauss diagrams. It deserves a serious referee. I’d send it out and ask for the missing proofs, a clarified matrix definition, and archived code/data.\n\nBest,","headline":"New relator framework for spherical-curve invariants; computational section is promising but needs a clearer match between the solved linear system and Theorem 1.","tokens_in":55737,"tokens_out":11926,"would_cite":false,"duration_ms":108867,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57M27","57M25"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that arrow-diagram counting functions become spherical-curve invariants when a delta-version vanishes on relators, and computes such invariants through six arrows.","keywords":["spherical curves","arrow diagrams","oriented Gauss words","Reidemeister moves","weak (1,3) homotopy","finite type invariants","normal oriented Gauss words"],"falsifier":"Run an independent enumeration of connected normal oriented Gauss words with five and six arrows and recompute the rank of the weak-RIII relator matrix; if the number of isomorphism classes in $\\check{G}_{5,6}\\cap\\mathrm{Conn}$ is not 15,922 or the kernel dimension is not 31, the paper's computational claim fails. For the three-arrow stage the matrix is printed in full, so the kernel can also be checked by hand.","tokens_in":54799,"feed_emoji":"➰","tokens_out":18862,"duration_ms":165311,"temperature":0.7,"pith_summary":"Spherical curves are generic circle immersions into the 2-sphere, studied up to five Reidemeister-style deformations: RI, strong and weak RII, and strong and weak RIII. The paper establishes a general criterion for turning arrow-diagram counting functions into integer-valued invariants under any chosen combination of those moves: one checks a much simpler delta-function version against a finite list of formal relators, and the check guarantees invariance. The authors then compute by enumerating normal oriented Gauss words and taking kernels of the resulting relator matrices. For weak (1,3) homotopy—RI together with weak RIII—the computation yields dimensions 1, 3, 13, 31 at successive arrow counts, including the explicit three-arrow invariant $-x^*_1+x^*_2-x^*_3-x^*_4-x^*_5-x^*_6+3x^*_7$. The motivation is an open question about whether the map taking a spherical curve to its positive knot diagram is injective under this homotopy.","feed_headline":"Six-arrow diagrams yield 31 spherical-curve invariants","feed_subtitle":"A relator-kernel computation proves invariance and counts weak (1,3) homotopy invariants up to six arrows.","key_machinery":"The key object is the arrow diagram, here identified with an oriented Gauss word: a cyclic word in which each letter occurs twice, once marked as a starting point and once as an end point. For each isomorphism class $x^*_i$, the function $x^*_i(AD)$ counts how many sub-arrow-diagrams of $AD$ are isomorphic to $x^*_i$, while $\\tilde{x}^*_i$ is the Kronecker delta on that single isomorphism class. The identity $x^*_i(AD)=\\sum_{z^*\\in\\mathrm{Sub}(G^*)}\\tilde{x}^*_i(z^*)$ reduces every Reidemeister-style deformation to a difference of such sums, and the relators are exactly those differences written as elements of the free $\\mathbb{Z}$-module on arrow diagrams. The computational engine is the enumeration of normal oriented Gauss words—words whose first occurrences of letters read $1,2,\\dots,n$ in order—by inserting the first letter into all possible positions and then lexicographically assigning orientations; the kernel of the resulting matrix $(\\tilde{x}^*_i(r^*_j))$ is the space of invariants.","core_discovery":"The central result (Theorem 1) is a transfer principle. Fix two arrow-count bounds $b,d$ and a subset of the five move types, encoded by $(\\epsilon_1,\\dots,\\epsilon_5)\\in\\{0,1\\}^5$, and form the finite set of relators $\\check{R}_{\\epsilon}(b,d)$ obtained by projecting the Type ($\\check{\\mathrm{I}}$), strong/weak ($\\check{\\mathrm{II}}$), and strong/weak ($\\check{\\mathrm{III}}$) relators to arrow diagrams with between $b$ and $d$ arrows. If a linear combination $\\sum\\alpha_i\\tilde{x}^*_i$ of delta functions on arrow-diagram isomorphism classes vanishes on every relator in the selected family, then the corresponding counting function $\\sum\\alpha_i x^*_i$ is an integer-valued invariant of oriented spherical curves under the corresponding deformations; a mirroring-pair condition makes the invariant independent of orientation. Computers then compute the kernel dimensions of the matrices $(\\tilde{x}^*_i(r^*_j))$. For weak (1,3) homotopy the dimensions are $\\dim V^{\\mathrm{ori}}_w(2,3)=1$, $\\dim V^{\\mathrm{ori}}_w(3,4)=3$, $\\dim V^{\\mathrm{ori}}_w(4,5)=13$, and $\\dim V^{\\mathrm{ori}}_w(5,6)=31$; the paper also lists the explicit coefficients $-x^*_1+x^*_2-x^*_3-x^*_4-x^*_5-x^*_6+3x^*_7$ at the three-arrow stage and notes, citing the finite-type invariant literature, that this particular one vanishes on spherical curves. For RI together with strong RIII the computed dimensions are 3, 18, 145.","pith_inferences":["The same relator-kernel computation can be run for any other subset of the five move types, including mixed combinations such as RI plus strong RII plus weak RIII, so the method is a general source of invariants for all 32 equivalence relations once the normal oriented Gauss word enumeration is available.","A direct next step is to evaluate the 13- and 31-dimensional invariants on the prime spherical curves with up to seven double points; if they take equal values on curves like 7_4 and 7_B, the filtration remains coarse at this order, while any separation would give the first computational evidence about the injectivity question that motivates the paper.","Because the relator matrices depend only on the inclusion poset of arrow diagrams, the same enumeration and matrix data can be reused when the move set changes, so the computational cost is amortized across many equivalence relations at once."],"forward_implications":["Any kernel vector of a relator matrix yields an integer-valued invariant, so the paper converts the search for invariants under any of the 32 equivalence relations generated by the five moves into a finite linear-algebra computation.","The weak (1,3) homotopy dimensions 1, 3, 13, 31 show new invariants appear at each successive arrow-count stage through six arrows; the paper conjectures these dimensions never exceed the corresponding knot-theoretic finite-type invariant dimensions.","Restricting to connected arrow diagrams (Corollary 2) makes the invariants additive under connected sum, so the computed invariants behave like finite-type invariants with respect to connect-sum decomposition.","The explicit three-arrow invariant, even though it vanishes on spherical curves according to the cited review, gives a pattern that the 13- and 31-dimensional spaces refine, and it provides a concrete check for any independent implementation."],"supporting_citations":[{"why":"Supplies the oriented Gauss word and Gauss diagram formalism, including the sub-word substitution identity the transfer argument relies on.","marker":"[16]"},{"why":"Provides the knot-theoretic Polyak algebra dimensions against which the computed weak (1,3) quotients are compared and that motivate Conjecture 1.","marker":"[2]"},{"why":"Gives the finite-type invariant framework and the review source stating that the explicit three-arrow invariant vanishes on spherical curves.","marker":"[3]"},{"why":"Introduces the sub-chord diagram invariant that the paper's strong-RIII invariants reproduce, anchoring the computational output.","marker":"[10]"},{"why":"Establishes that the five move types generate 32 equivalence relations, fixing the deformation landscape the invariants are defined on.","marker":"[11]"},{"why":"Poses Question 1 on injectivity of the positive-knot map, the motivating problem for constructing these invariants.","marker":"[6]"}],"fun_headline_variants":["Arrow diagrams compute 31 invariants for spherical curves","31 spherical-curve invariants from arrow diagrams","Six-arrow limit: 31 new spherical curve invariants","Computing 31 invariants of spherical curves via arrows"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The computed dimensions and coefficient tuples rest on the completeness and duplicate-freeness of the enumeration of normal oriented Gauss words in Section 4; the paper spells out the procedure but gives no formal proof or independent audit of the enumeration, and the raw lists for the five- and six-arrow cases are not printed in the paper.","fun_headline_variants_meta":{"raw":{"variants":["Arrow diagrams compute 31 invariants for spherical curves","31 spherical-curve invariants from arrow diagrams","Six-arrow limit: 31 new spherical curve invariants","Computing 31 invariants of spherical curves via arrows"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000624,"raw_usage":{"total_tokens":3042,"prompt_tokens":1248,"completion_tokens":1794,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":864,"completion_tokens_details":{"reasoning_tokens":1729}},"tokens_in":864,"tokens_out":1794,"duration_ms":12155,"temperature":1.0,"reasoning_tokens":1729,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:03:10.877064+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run an independent enumeration of connected normal oriented Gauss words with five and six arrows and recompute the rank of the weak-RIII relator matrix; if the number of isomorphism classes in $\\check{G}_{5,6}\\cap\\mathrm{Conn}$ is not 15,922 or the kernel dimension is not 31, the paper's computational claim fails. For the three-arrow stage the matrix is printed in full, so the kernel can also be checked by hand.","supporting_citations":[{"cited_title":"Polyak and O","cited_arxiv_id":null,"evidence_quote":"Supplies the oriented Gauss word and Gauss diagram formalism, including the sub-word substitution identity the transfer argument relies on."},{"cited_title":"Bar-Natan, I","cited_arxiv_id":null,"evidence_quote":"Provides the knot-theoretic Polyak algebra dimensions against which the computed weak (1,3) quotients are compared and that motivate Conjecture 1."},{"cited_title":"Goussarov, M","cited_arxiv_id":null,"evidence_quote":"Gives the finite-type invariant framework and the review source stating that the explicit three-arrow invariant vanishes on spherical curves."},{"cited_title":"Ito and Y","cited_arxiv_id":null,"evidence_quote":"Introduces the sub-chord diagram invariant that the paper's strong-RIII invariants reproduce, anchoring the computational output."},{"cited_title":"Ito and Y","cited_arxiv_id":null,"evidence_quote":"Establishes that the five move types generate 32 equivalence relations, fixing the deformation landscape the invariants are defined on."},{"cited_title":"Ito, Knot projections, CRC Press, Boca Raton, FL, 2016","cited_arxiv_id":null,"evidence_quote":"Poses Question 1 on injectivity of the positive-knot map, the motivating problem for constructing these invariants."}],"review_version":1}