{"id":"92652a74-5738-4147-8874-86bf951fda6b","arxiv_id":"1908.01693","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"All 1655 alternating 11- and 12-crossing knots now have known tunnel numbers, with 142 non-alternating knots also determined.","lead":"This paper computes the tunnel number for every alternating knot with 11 or 12 crossings, and for 142 non-alternating knots in the same range. The table is built by applying a known classification of tunnel-number-one alternating knots to an exhaustive enumeration of the knots.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.1's rational-tangle enumeration omits fractions like ±1/2 because it uses positive continued fractions, conflicting with Definition 2.1's alternating signs; this could invalidate the Montesinos/clasp checks underlying Theorem 1.1.","rationale":"The reader's weakest assumption concerns e=0 in Theorem 4.2. I do not think that is the strongest failure mode: an integer e can be absorbed into an adjacent rational tangle (e + p/q has denominator q), and the crossing count of that tangle changes by at most |e|, so every minimal-crossing Montesinos presentation should have an e=0 representative with the same crossing number. The stated algorithm for rational tangles, however, is internally inconsistent with the paper's sign convention and has no exhaustiveness proof. This is a direct threat to Theorem 1.1, and it is checkable against the GitHub code. Because the computation may still be correct if the code differs from the text, the appropriate verdict remains conditional rather than accept or reject.","tokens_in":6598,"tokens_out":28759,"duration_ms":293344,"concrete_test":"Run the exact step (2) of Theorem 4.1 for ℓ = 2, 3, 4 and compare the fractions produced with the complete set of rational tangles with ℓ crossings, generated either from SnapPy's RationalTangle code or from all Conway sequences with Σ|a_i| = ℓ under the alternating-sign convention. Check specifically for ±1/2 and ±3/2; if any are missing, the enumeration is incomplete. Then rerun the Montesinos construction of Theorem 4.2 with a corrected RT list and compare the resulting 11- and 12-crossing alternating knots against the paper's Tables 1 and 2; any newly appearing clasp Montesinos knot would require recomputing its tunnel number by Lackenby's theorem.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.1 is the foundation of the Montesinos identification, but its step (2) is inconsistent with the paper's own Definition 2.1 and is not exhaustive as written. Definition 2.1 defines the fraction of a rational tangle by β/α = [a1, -a2, a3, ..., ±am], a continued fraction with alternating signs; under this convention the clasp tangle ±1/2 is represented by [1,-2] (equivalently [0,2] with a zero first entry). Theorem 4.1 instead forms p/q = a1 + 1/(a2 + ...) from partitions of ℓ into positive integers and then adds only the negatives. For ℓ=2 this yields {±2}; for ℓ=3 it yields fractions such as 3/2 but not 1/2. Thus RT(ℓ) can miss the very clasp tangles on which Lackenby's classification and Proposition 5.1 Case 2 depend. The proof's one-line justification ('every alternating diagram is a minimal diagram') does not bridge the gap between the displayed continued-fraction formula and the required enumeration of all fractions with an alternating continued fraction of length ℓ. If the submitted code reproduces Theorem 4.1 literally, the subsequent list of Montesinos knots (Theorem 4.3) is incomplete and the tunnel numbers assigned in Propositions 5.1–5.2—especially the distinction between tunnel number one and two for 3-bridge knots—could be wrong.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to compute the tunnel number of all 1655 alternating 11- and 12-crossing knots, and of 142 non-alternating 11- and 12-crossing knots, by combining Lackenby's classification of tunnel-number-one alternating knots, Boileau–Zieschang bridge numbers for Montesinos knots, Lustig–Moriah's theorem for Montesinos knots with nontrivial gcd of denominators, and KnotInfo bridge-number data. The central tool is an enumeration of all Montesinos knots with at most 14 crossings, based on partitioning the crossing number among rational tangles and using SnapPy for identification. The main theorem then assigns tunnel numbers case by case according to bridge number and Montesinos/clasp status.","tokens_in":6896,"tokens_out":11437,"duration_ms":112178,"significance":"If correct, the paper would supply the first complete tunnel-number table for alternating knots with 11 and 12 crossings, a useful data resource for testing conjectures about tunnel number. The methodology is attractive: it relies on established theorems (Lackenby, Lustig–Moriah, Boileau–Zieschang), external bridge-number data from KnotInfo, and reproducible code on GitHub, with no fitted parameters or circular dependencies. The case split in Proposition 5.1 is natural once the Montesinos classification is trusted. However, the exhaustiveness of the Montesinos enumeration is not established as written, and the manuscript's own examples use fractions that the stated algorithm cannot generate; these gaps are load-bearing for the main theorem.","major_comments":[{"comment":"The enumeration in Theorem 4.1 is incomplete. It forms fractions from partitions of ℓ into positive integers using the positive continued fraction p/q = a1 + 1/(a2 + ... ), and then adds negatives. But rational tangles with fraction of absolute value less than 1, such as 1/2, 2/3, and 1/3, have standard continued fractions with a zero first entry, e.g. 1/2 = [0,2] and 2/3 = [0,1,2]. Such fractions are not produced by partitions into positive integers. For ℓ=2, RT(2) contains only 2 and -2, not ±1/2; for ℓ=3, RT(3) contains 3 and 3/2, not ±1/3 or ±2/3. The one-line justification that every alternating diagram is minimal does not address the zero-coefficient issue. This is not merely a proof gap: Proposition 5.1 explicitly names the Montesinos knots M(0; 2/3, 2/3, 2/3, 1/3) and M(0; 2/3, 1/3, 1/3, 1/3), whose fractions 2/3 and 1/3 are absent from RT(3) as defined. Hence the stated algorithm cannot be the one that produced the list behind Theorem 4.3, and the described enumeration would miss exactly the clasp tangles on which the tunnel-number classification depends.","section":"Theorem 4.1"},{"comment":"Theorem 4.2 does not account for the e parameter in the Montesinos notation. The algorithm writes M(e; p1/q1, ..., pr/qr) but never specifies what integer values of e are considered, and the partitions of n into the tangle crossing numbers leave no room for the |e| crossings of the extra twist region. The proof does not show that every Montesinos knot with n crossings admits a presentation with e=0 and with rational tangles whose crossing numbers sum to n. Since Lackenby's classification in Theorem 3.6 explicitly includes the parameter e in the clasp Montesinos form M(e; ±1/2, β1/α1, β2/α2), a nontrivial e can arise in the very class the paper needs to enumerate completely. Without an argument covering e, the exhaustiveness claim in Theorem 4.3 is unsupported.","section":"Theorem 4.2"},{"comment":"In Case 3 of Proposition 5.1, the sentence 'Every knot in this case is a Montesinos knot' is asserted without proof or citation. This is not a trivial fact: alternating 4-bridge knots are not in general Montesinos. The paper gives no theorem or detailed SnapPy procedure establishing that, among the 1655 alternating knots with 11 or 12 crossings, every bridge-number-4 example is Montesinos. If the Montesinos list generated by Theorems 4.1–4.3 is incomplete, this assertion and the identification of 12a0554 and 12a0750 as the only two exceptions are not reliable. The proof needs either a rigorous argument or a precise description of how the exhaustive SnapPy identification was performed and why it is complete.","section":"Proposition 5.1, Case 3"}],"minor_comments":[{"comment":"The abstract states that the paper computes the tunnel number of '881 non-alternating 11 and 12 crossing knots', while Theorem 1.1 and the body consistently report 142 non-alternating knots. This mismatch should be corrected.","section":"Abstract"},{"comment":"There are many typographical errors, for example 'al ternating' in the title, 'm any' in the abstract, and 'kn ots' in Theorem 1.1. A careful proofreading pass is needed.","section":"Throughout"},{"comment":"The formatting in Table 1 is difficult to parse, for instance the line 'Non-clasp: 37 α = 1: 35 Tunnel number 2: 37α ⁄= 1: 2' runs multiple entries together without clear separation. The table should be restructured so that each count (α=1, α≠1, tunnel number) is visually distinct.","section":"Table 1"},{"comment":"The phrase 'construct a Montesinos link by tangle summing together all rational tangles from the tuple in order, and then taking the numerator closure of the sum' is ambiguous, and the formula 'M(e; p1/q1, ..., pr/qr)' introduces e without defining how e is chosen. The construction should be stated precisely, including the role of e and the crossing count of the resulting diagram.","section":"Theorem 4.2, step (3)"}],"recommendation":"major_revision","confidential_remarks":"The GitHub repository is a positive feature and may allow the authors to document exactly what enumeration was actually implemented. The gap between the stated algorithm and the examples in Proposition 5.1 suggests the written theorem does not match the code; if the code contains the missing fractions (e.g. via SnapPy's RationalTangle with zero entries) and handles e, the results may be correct, but the manuscript must be rewritten to describe the true algorithm and prove its completeness. Otherwise the main theorem is unsupported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nYou should know two things about arXiv:1908.01693. First, it fills a genuine gap: nobody had computed tunnel numbers for all alternating 11- and 12-crossing knots. Second, the main computational step is flawed as written, and the flaw is load-bearing.\n\nTheorem 4.1 claims to enumerate all rational tangles with ℓ crossings by taking integer partitions of ℓ and forming positive continued fractions. But Definition 2.1 uses alternating-sign continued fractions, and the clasp tangle ±1/2 never appears. For ℓ=2, the algorithm produces ±2 only. Lackenby's classification says an alternating knot has tunnel number one exactly if it is 2-bridge or a clasp Montesinos knot with a ±1/2 tangle. If the enumeration misses that tangle, the Montesinos list is incomplete, and Proposition 5.1 will assign tunnel number two to every alternating 3-bridge clasp Montesinos knot instead of one. That is not a minor edge case; it likely flips the counts for a large subset of the table.\n\nThe paper does have strengths. The structure is clear, the reliance on Lackenby and Lustig-Moriah is appropriate, and the summary tables are useful. The case analysis in Propositions 5.1 and 5.2 is logical once you accept the lists. The authors also put code and data on GitHub, which is good practice.\n\nThere are two additional soft spots. Theorem 4.2 does not specify how the parameter e in M(e; ...) is handled, so the Montesinos enumeration may also miss knots with e≠0. And the abstract advertises 881 non-alternating tunnel numbers and 5525 Montesinos knots with 14 or fewer crossings, while the body reports 142 and only gives 11/12 counts. That discrepancy needs a fix.\n\nIf the GitHub code actually includes the clasp tangles (e.g., by also adding fractions of the form ±1/k or by using SnapPy's own tangle enumeration), the final numbers might be correct. But the paper as written does not demonstrate that. I would not trust the tables without seeing the code output.\n\nMy recommendation: send it to peer review, but clearly tell the authors to fix the enumeration, specify the e-parameter handling, and reconcile the abstract with the body. The project is worth doing; this version is not yet reliable.\n\nBest","headline":"The tunnel-number table is built on an enumeration that omits ±1/2, so the main result is unsupported unless the code does more than the paper says.","tokens_in":7431,"tokens_out":10296,"would_cite":false,"duration_ms":96347,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57M25","57M27"],"pacs":[],"model":"deepseek-v4-flash","headline":"The tunnel number of every 11- or 12-crossing alternating knot has been computed, along with 142 non-alternating knots.","keywords":["tunnel number","alternating knots","Montesinos knots","rational tangles","bridge number","knot enumeration","low-crossing knots"],"falsifier":"Run a computer search over all integers e and all rational tangles whose crossing counts sum to 11 or 12, construct the Montesinos knots M(e; ...), and check whether any resulting knot is absent from the paper's list; even one such knot would break the claimed exhaustiveness and could change the tunnel number assigned to it.","tokens_in":6347,"feed_emoji":"🪢","tokens_out":8172,"duration_ms":72499,"temperature":0.7,"pith_summary":"This paper completes the tunnel number table for alternating knots with 11 and 12 crossings, and adds exact values for 142 non-alternating knots. It does this by combining a known classification of alternating tunnel-number-one knots, a theorem giving tunnel number for certain Montesinos knots whose tangle denominators share a divisor, and an exhaustive enumeration of all Montesinos knots in this range. If the calculations are right, every alternating knot with 11 or 12 crossings is assigned tunnel number 1, 2, or 3, with exactly two 12-crossing knots attaining 3. The enumeration also yields all Montesinos knots with 14 or fewer crossings, counted as 5525 knots.","feed_headline":"Tunnel numbers for all 1,655 alternating 11- and 12-crossing knots","feed_subtitle":"Every value is 1, 2, or 3; only two 12-crossing knots reach 3.","key_machinery":"The load-bearing object is the Montesinos knot $M(e; \\beta_1/\\alpha_1, \\ldots, \\beta_r/\\alpha_r)$, built by summing $r$ rational tangles and taking the numerator closure; the clasp case has exactly one tangle equal to $1/2$. The mechanism is a chain: an algorithm enumerates all rational tangles with $\\ell$ crossings as continued-fraction values of integer partitions of $\\ell$, then constructs all Montesinos knots with $n$ crossings by partitioning $n$ among $3 \\le r$ tangles; a classification theorem says an alternating knot has tunnel number one exactly when it is a 2-bridge knot or a clasp Montesinos knot with three tangles; and a separate theorem gives $t(K) = r - 1$ when the tangle denominators share a nontrivial common divisor. Bridge number data, at most 4 for all knots in this range, chooses which theorem applies.","core_discovery":"The paper's central claim is Theorem 1.1: the tunnel number of all 1655 alternating 11- or 12-crossing knots has been calculated, along with the tunnel number of 142 non-alternating knots in the same crossing range. The method is exhaustive: every such knot is sorted by bridge number, which is at most 4 in this range, and for 3- and 4-bridge knots the paper identifies whether it is a clasp Montesinos knot using an algorithm that constructs every Montesinos knot with n crossings from integer partitions and rational-tangle fractions. Known theorems then determine the tunnel number: 2-bridge knots have tunnel number 1; alternating 3-bridge knots have tunnel number 1 if clasp Montesinos and 2 otherwise; alternating 4-bridge clasp Montesinos knots have tunnel number 2; and the two remaining 12-crossing alternating knots are non-clasp Montesinos knots whose tangle denominators share a divisor, giving tunnel number 3.","pith_inferences":["If the algorithm's exhaustiveness holds, the same partition-based recipe could be run at 13 and 14 crossings; the main obstacle is not the classification theorems but the size of the rational-tangle enumeration and the reliability of knot identification.","The paper leaves only two alternating 12-crossing knots at tunnel number 3; a natural next test is whether this sparsity persists at 13 crossings or whether higher tunnel numbers begin to appear among alternating knots.","A direct check of the e=0 assumption would be to search for Montesinos diagrams with a nonzero integer twist whose total crossing count is 11 or 12 and compare the resulting knots against the paper's list; this would settle the enumeration's completeness without relying on computer identification.","The method's dependence on bridge number data implies that any larger alternating knot with known bridge number at most 4 would automatically receive a tunnel number, making enumeration the limiting step for further extension."],"forward_implications":["Every alternating 11- or 12-crossing knot now has tunnel number 1, 2, or 3, with 1 and 2 covering all but the two knots 12a0554 and 12a0750.","The table of 1797 exact tunnel numbers provides a concrete data set for testing conjectures that relate tunnel number to hyperbolic volume, bridge number, or connected sums.","The exhaustive enumeration identifies all 5525 Montesinos knots with 14 or fewer crossings, which is exactly the family needed to apply the classification theorem in this range.","For the 931 non-alternating 11- and 12-crossing knots whose tunnel number is not fixed, the paper gives the bound that each has tunnel number 1 or 2.","The method extends the prior complete table for knots with at most 10 crossings to the alternating 11- and 12-crossing range."],"supporting_citations":[{"why":"Supplies the classification of alternating tunnel-number-one knots as 2-bridge or clasp Montesinos knots, the backbone of the tunnel-number determinations.","marker":"[7]"},{"why":"Gives $t(K) = r - 1$ for Montesinos knots whose tangle denominators share a nontrivial divisor, used for the two 12-crossing alternating knots and for non-alternating cases.","marker":"[8]"},{"why":"Establishes that a Montesinos knot with $r$ rational tangles has bridge number $r$, letting the paper reduce bridge-number cases to tangle counting.","marker":"[1]"},{"why":"Supplies bridge numbers for all 11- and 12-crossing knots, the input that selects which case applies.","marker":"[3]"},{"why":"Provides the knot identification used to match constructed Montesinos knots to the standard knot tables.","marker":"[5]"},{"why":"Gives the previous complete tunnel-number table through 10 crossings, the baseline this paper extends.","marker":"[12]"}],"fun_headline_variants":["Exhaustive computation yields tunnel numbers for all 1,655 alternating knots","Tunnel number found for every alternating 11- and 12-crossing knot","All alternating 11- and 12-crossing knots have tunnel number 1, 2, or 3","Only two 12-crossing alternating knots have tunnel number 3","1,655 alternating knots: complete tunnel number determination"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole table rests on the assumption that the algorithm that partitions the 11 or 12 crossings only among the rational tangles, with the extra integer twist parameter set to zero, catches every Montesinos knot in this range.","fun_headline_variants_meta":{"raw":{"variants":["Exhaustive computation yields tunnel numbers for all 1,655 alternating knots","Tunnel number found for every alternating 11- and 12-crossing knot","All alternating 11- and 12-crossing knots have tunnel number 1, 2, or 3","Only two 12-crossing alternating knots have tunnel number 3","1,655 alternating knots: complete tunnel number determination"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000822,"raw_usage":{"total_tokens":3520,"prompt_tokens":789,"completion_tokens":2731,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":405,"completion_tokens_details":{"reasoning_tokens":2628}},"tokens_in":405,"tokens_out":2731,"duration_ms":20782,"temperature":1.0,"reasoning_tokens":2628,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:06:54.676344+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a computer search over all integers e and all rational tangles whose crossing counts sum to 11 or 12, construct the Montesinos knots M(e; ...), and check whether any resulting knot is absent from the paper's list; even one such knot would break the claimed exhaustiveness and could change the tunnel number assigned to it.","supporting_citations":[{"cited_title":"Lackenby, Classification of alternating knots with tunnel number one","cited_arxiv_id":null,"evidence_quote":"Supplies the classification of alternating tunnel-number-one knots as 2-bridge or clasp Montesinos knots, the backbone of the tunnel-number determinations."},{"cited_title":"Lustig, Y","cited_arxiv_id":null,"evidence_quote":"Gives $t(K) = r - 1$ for Montesinos knots whose tangle denominators share a nontrivial divisor, used for the two 12-crossing alternating knots and for non-alternating cases."},{"cited_title":"Boileau and H","cited_arxiv_id":null,"evidence_quote":"Establishes that a Montesinos knot with $r$ rational tangles has bridge number $r$, letting the paper reduce bridge-number cases to tangle counting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies bridge numbers for all 11- and 12-crossing knots, the input that selects which case applies."},{"cited_title":"Culler, N","cited_arxiv_id":null,"evidence_quote":"Provides the knot identification used to match constructed Montesinos knots to the standard knot tables."},{"cited_title":"Morimoto, M","cited_arxiv_id":null,"evidence_quote":"Gives the previous complete tunnel-number table through 10 crossings, the baseline this paper extends."}],"review_version":1}