Knot Structures — Preprints

The Apprentice Folder (workshop kit): a working procedure for an AI tool in an existing environment

2026-11-06T00:00:00Z

The Apprentice Folder ("workshop kit") is the working procedure of the project Knot Structures, packed as a folder to take away: a boot order, a working diary, two growing knowledge files (numbered lessons, an error log), a map, four shell gates and a closing routine, for an AI tool started inside an existing environment. Its one load-bearing rule: at first the AI only observes and writes, it changes nothing; a measuring script, not a calendar, decides when that phase ends. The rule is a recommendation, not a lock. Evidence status: nine trial runs (August to 5 November 2026), all in our own e

θ detects chirality for all prime knots up to 14 crossings — and first fails at 15: the chiral θ-trivial knots 15nh_048139 and 15nh_048140

2026-11-04T00:00:00Z

Θ = (Δ, θ) is the knot invariant of Bar-Natan and van der Veen (arXiv:2509.18456v4). Their Conjectures 21 and 22 give "amphichiral ⟹ θ = 0". This archive measures the converse over the complete census of prime knots up to 15 crossings (313,230 knots, Burton's tables; counts equal OEIS A002863 exactly; 0 errors). RESULTS: (1) θ(mirror K) = −θ(K) at the point (22/7, 21/13) for every one of the 313,230 knots. (2) For c ≤ 14 (59,937 knots) θ = 0 at the point holds for exactly 352 knots, and each is CERTIFIED amphichiral (351 by SnapPy verified oriented isometry signatures, one, 14ah_00990, by an e

OEIS A343358 extended: a(14) = 17,831 and a(15) = 74,581 connected realizable circle graphs, with the graphs themselves

2026-11-03T00:00:00Z

OEIS A343358 counts the connected graphs on n vertices that are realizable as circle (chord-interlacement) graphs of Gauss diagrams of closed planar curves with n crossings (Vernitski; Khan–Lisitsa–Lopatkin–Vernitski, arXiv:2108.02873v1); known terms n = 3..13, keyword hard. This archive gives the next two terms, a(14) = 17,831 and a(15) = 74,581, and the complete lists of the graphs for n = 13, 14, 15 in canonical graph6 form, each with the Burton-census names of the prime alternating knots whose minimal diagram realizes it (4,573 / 17,831 / 74,581 data lines). REDUCTION: a connected realizab

The positive 3-braid knots σ₁³σ₂³σ₁⁶σ₂⁶ and σ₁³σ₂⁵σ₁³σ₂⁷ are not concordant

2026-10-30T00:00:00Z

The closures K1 of σ₁³σ₂³σ₁⁶σ₂⁶ and K2 of σ₁³σ₂⁵σ₁³σ₂⁷ are two non-isotopic positive 3-braid knots with 18 crossings that P. Truöl lists as a pair she could not distinguish in concordance (Oberwolfach Report 43/2024, DOI 10.4171/OWR/2024/43, Question 4: "Are concordant braid positive knots isotopic?"). RESULT: K1 and K2 are not topologically concordant, hence not smoothly concordant. METHOD: the Kirk–Livingston / Herald–Kirk–Livingston twisted Alexander polynomial test (a Casson–Gordon-type invariant), in the form of Theorem "main KL" of Dunfield–Gong, arXiv:2512.21825v1, applied to K1 # −K2 w

knot-identification-pipeline: knot type of closed space curves from their geometry

2026-11-07T00:00:00Z

Identifies the knot type of a closed space curve given as a Fourier series, without consulting knot tables during identification: GPU camera-flight crossing scanner, crossing tracker, diagram resolution, simplification (Regina, spherogram), invariants (SnapPy, SageMath, KnotJob, Bar-Natan–van der Veen Theta), and only afterwards a comparison with KnotInfo, Fremlin and the SnapPy census. Built for the stainless-steel knot sculptures of Werner Alois Stanggassinger (sculptor). Werner Alois Stanggassinger in collaboration with Claude Code (Anthropic): the choice of problem, the direction of the wo

Turk's head knots with at least three leads and three bights are not perfectly straight

2026-10-26T00:00:00Z

Owad defined the straight number str(K) of a knot, the least number of crossings of a diagram in which one straight strand meets all crossings, and called K perfectly straight if str(K) equals the crossing number. Di Prisa and Savk asked whether the Turk's head knot Th(p,q) is perfectly straight when p >= 3 and q < p (Problem A of their survey, arXiv:2409.20106). We prove that Th(p,q) is not perfectly straight for all p, q >= 3 with gcd(p,q) = 1. For q < p this answers Problem A when q >= 3; for q > p it is a theorem of Owad, of which we give an independent proof. The proof combines an exact c

Positive amphichirality of Turk's head knots: the orders of the maps

2026-10-26T00:00:00Z

Let p >= 3 be odd, q >= 3 and gcd(p,q) = 1. We show that the Turk's head knot Th(p,q) is strongly positive amphichiral if and only if q is odd, and that the orders of the finite-order diffeomorphisms realising its positive amphichirality are exactly the numbers 2q/d, d an odd divisor of q. This answers Problem D of Di Prisa and Savk (arXiv:2409.20106) for these knots. The proof reads the answer off the symmetry group D_2q determined in a companion paper (arXiv:2610.16600), together with the action of each symmetry on the orientation of the knot. For q = 2 and p >= 5 odd we show that the two po

Grid diagrams of nontrivial knots: a(9) and a(10) of OEIS A163566

2026-10-26T00:00:00Z

Two new terms of OEIS A163566, "Total number of grid diagrams of nontrivial knots on grid size n" (Baldridge and McCarty, arXiv:0907.5401v1): a(9) = 873,994,338 and a(10) = 109,051,496,360, both computed exactly. The OEIS entry (revision 8) has the four terms n = 5..8; our code reproduces all four. METHOD: every grid diagram of size n (pair of permutations X, O with X(i) != O(i) whose induced permutation is one n-cycle) is enumerated on one GPU (CuPy, RTX 3070). The Alexander polynomial is tested through the minesweeper determinant det M(G) = +-t^k (1-t)^(n-1) Delta(t) (Manolescu, Ozsvath, Sza

A certified census of spiral knots: no satellites and no coincidences in a finite range

2026-10-26T00:00:00Z

The spiral knot S(p,q,eps) is the closure of the braid (sigma_1^eps_1 ... sigma_{p-1}^eps_{p-1})^q. Blackwell, Das, Mayer, Moyar, Quraishi and Stees (arXiv:2506.17889) asked whether spiral knots can be satellite knots (Question 5.1) and when two spiral knots coincide (Question 5.2), and conjectured that two equal spiral knots have the same parameters (p,q) unless they are torus knots (Conjecture 5.5). We check all three in the range of coprime p, q with 3 <= p <= 16 and braid length q(p-1) <= 55, together with all coprime 3 <= p <= 9, 2 <= q <= 11: 20,428 parameter triples up to the two symmet

Two of Jabłonowski's candidate inequalities hold for all prime knots with at most 18 crossings

2026-11-03T00:00:00Z

Jabłonowski (arXiv:2605.22652) proposed nine candidate inequalities between integer knot invariants and proved structural special cases. Two of them compare the concordance invariants of Ozsváth–Szabó and Rasmussen with the z-degree of the HOMFLY polynomial: 2|tau(K)| <= deg_z P(K) and |s(K)| <= deg_z P(K). We show by an exact computation that both hold for every prime knot with at most 18 crossings, and hence for every connected sum of such knots. The genus and tau come from knot Floer homology (Szabó's program in SnapPy), s (where it is needed) from Schütz's KnotJob, and deg_z P from Regina,

The symmetry group of the Turk's head knots Th(p,2)

2026-10-22T00:00:00Z

Di Prisa and Savk conjectured that the Turk's head knot Th(p,q), gcd(p,q) = 1, has symmetry group the dihedral group of order 4q (Conjecture R of their survey, arXiv:2409.20106). The case q >= 3 is proved in arXiv:2610.16600, which excludes q = 2 because the standard diagram has bigons. This note treats q = 2: for every odd p >= 3, Th(p,2) is a hyperbolic, amphicheiral two-bridge knot, and hence its symmetry group is the dihedral group of order 8, as conjectured. The proof combines three known facts with one observation, that Th(p,2) is a two-bridge knot; for p = 3 (the figure-eight knot) the

Prime knots with twisted Alexander vanishing order 24 up to 19 crossings

2026-10-22T00:00:00Z

Ishikawa, Morifuji and Suzuki defined the twisted Alexander vanishing (TAV) order O(K) of a knot, showed O(K) >= 24, with equality exactly when the symmetric group S4 is a TAV group of K; among the prime knots with at most ten crossings, O(K) = 24 holds exactly for 9_35 and 9_46, as shown by Morifuji and Suzuki. In the sequel (arXiv:2510.25574v1, p. 3) they ask how O(K) behaves on special classes of knots, among them knots with a given crossing number. We decide O(K) = 24 for each of the 352,152,252 prime knots with at most 19 crossings in Burton's census: it holds for exactly 18,436 of them.

The symmetry group of Turk's head knots

2026-10-26T00:00:00Z

For p, q >= 3 with gcd(p,q) = 1 we show that the symmetry group of the Turk's head knot Th(p,q), the closure of the braid (sigma_1 sigma_2^{-1} sigma_3 sigma_4^{-1} ... sigma_{p-1}^{+-1})^q, is the dihedral group of order 4q. This proves Conjecture R of Di Prisa and Savk for q >= 3; the case p = 3 is due to Sakuma and Weeks. The argument is diagrammatic: by the flyping theorem of Menasco and Thistlethwaite, the symmetries of a knot with a prime, reduced alternating diagram without non-trivial flypes are the symmetries of the diagram (we write out a proof for the diagrams at hand), and we show

Conway polynomials of Th(p,q) and Th(q,p): a proof of Takemura's conjecture

2026-10-22T00:00:00Z

Takemura conjectured that the Conway polynomials sum a_l z^l of the Turk's head link Th(p,q) and sum b_l z^l of Th(q,p) satisfy a_l = b_l for l = 0,1 (mod 4) and a_l = -b_l for l = 2,3 (mod 4), and proved it for q = 2, 3. We prove it for all p, q >= 1, knots and links. Equivalently, nabla_Th(q,p)(z) = c nabla_Th(p,q)(iz) with c = 1 if gcd(p,q) is odd and c = -i if it is even. The proof combines a product formula nabla_Th(p,q)(z)^2 = prod_{j=1}^{p-1} prod_{k=1}^{q-1} (z^2 + 2cos(2 pi k/q) - 2cos(2 pi j/p)), which comes from the reduced Burau representation, with the leading coefficients, read o

Vol-Det, Fox, Hoste and a conjecture of Hirasawa–Murasugi for all prime alternating knots with at most 19 crossings

2026-10-22T00:00:00Z

For every prime alternating knot with at most 19 crossings in B. A. Burton's census (51 280 976 knots, of which 51 280 967 are hyperbolic) we verify: the Vol-Det conjecture of Champanerkar, Kofman and Purcell, vol(K) < 2π log det(K) (for the hyperbolic ones), published for up to 16 crossings; Fox's trapezoidal conjecture in its strict form; log-concavity of the Alexander coefficients; Hoste's conjecture that every zero of the Alexander polynomial has real part > -1; and Conjecture 5.11 of Hirasawa and Murasugi on the width of the plateau. The verification is done knot by knot: Vol-Det by Adams

The degree of the Alexander polynomial of twisted torus knots for every s

2026-11-04T00:00:00Z

For the twisted torus knot K(p,q;r,s), the closure of delta_p^q delta_r^(rs) with 0 < q < p, gcd(p,q) = 1, 2 <= r < p and s != 0, we give a closed formula for the degree (highest minus lowest exponent) of the Alexander polynomial, valid for every s. It is expressed through the degree sequence (d_m) of our earlier paper on the leading coefficient: deg Delta = 2 max_m (d_m + m r s) - (q+1)(p-1) - (r-1) r s. For s < 0, Park and Adnan determined it in two regimes. The highest exponent comes from the earlier paper, whose two theorems used here are re-proved in an appendix; the lowest follows from t

Hoste's condition for all Turk's head knots

2026-10-22T00:00:00Z

We show that every zero z of the Alexander polynomial of every Turk's head knot Th(p,q) satisfies Re z > -1, and that every non-real zero lies on the unit circle. This answers Problem H of Di Prisa and Savk, which asks to verify Hoste's condition for all Turk's head knots; the survey records it for p = 3. The proof is an exact factorisation of det(I - x S(sigma_1 sigma_2^-1 sigma_3 ... sigma_{p-1}^{+-1})) for the reduced Burau representation S into floor((p-1)/2) quadratic factors in t, whose middle coefficients involve c_j = 4cos^2(pi j/p), and possibly one linear factor, followed by an eleme

The Alexander plateau of a two-bridge knot is carried by one coherent twist region

2026-10-22T00:00:00Z

Let K be a two-bridge knot whose Alexander polynomial has a central plateau of half-width m >= 1, i.e. 2m+1 equal central coefficients in absolute value. We show that the reduced alternating (Conway) diagram of K has a coherently oriented twist region in which a single crossing change produces a two-bridge knot K' with c(K') = c(K) - 2, plateau half-width m - 1 and |sigma(K')| = |sigma(K)| - 2, except for the trefoil, where K' is the unknot. This is a converse-type statement, for two-bridge knots, to the twist step of Azarpendar, Juhasz and Kalman (arXiv:2406.08662), and it shows that the plat

Boninger's Conjectures 1.1, 1.6 and 1.7 hold for all prime alternating knots with at most 19 crossings: a computer check

2026-10-22T00:00:00Z

J. Boninger, 'An Alexander polynomial refinement for alternating links, with trapezoidal properties' (arXiv:2608.28484v1, 28 August 2026), states Conjectures 1.1 (symmetry P(x,y,z,w) = P(y,x,w,z)), 1.6 (M-convex support) and 1.7 (log-concavity) and reports them verified for all alternating knots with at most eleven crossings. We checked all three on every alternating knot in B. A. Burton's census of prime knots up to 19 crossings: 51 280 976 knots (51 280 967 hyperbolic and the nine torus knots T(2,n), n = 3, ..., 19), of which 51 280 413 have 12 to 19 crossings. No counterexample was found. T

The Andersen–Kashaev volume conjecture for 220 knots beyond the 14-crossing census of Ben Aribi, Guilloux and Wong

2026-10-22T00:00:00Z

For 220 knots beyond the 14-crossing census of Ben Aribi, Guilloux and Wong we exhibit an ordered ideal triangulation of the knot complement that is FAMED and geometric; by Ben Aribi-Wong, Corollary 1.9 (arXiv:2410.10776v2: 'Conjecture 1.5 is true for all FAMED geometric triangulations'), the Andersen-Kashaev volume conjecture holds for these knots. 216 distinct complements (4 chiral mirror pairs). THE THEOREM AND THE METHOD ARE NOT OURS: the conjecture is Andersen-Kashaev's (arXiv:1109.6295), FAMED triangulations and Corollary 1.9 are Ben Aribi-Wong's, the census-wide computer proof and the s

Symmetric ribbon numbers: r_s(12n706) = 4, a candidate for 12n504, and a computed subset of R^s_5

2026-10-22T00:00:00Z

Version 2 (17 October 2026). One symmetric ribbon number determined by an explicit witness, r_s(12n706) = 4; a candidate upper bound for 12n504 that the source's Lemma 2.3 does not cover as written; and a computed subset of the set R^s_5. NO THEOREM OF OURS IS CLAIMED AND NONE OF THE MATHEMATICS IS OURS: the notion of a symmetric ribbon number, the lemmas used, the bounds and the tables belong to Akash, Corrado, Placke, Sanketh, Starns, Timothy and Zupan, 'Symmetric ribbon numbers of low-complexity knots', arXiv:2606.02390v1 (1 June 2026; still v1 when checked on 17 October 2026). What is depo

The leading coefficient of the Alexander polynomial of twisted torus knots

2026-11-06T00:00:00Z

For a twisted torus knot K(p,q;r,s) with gcd(p,q) = 1 and 2 ≤ r < p we determine the leading coefficient of the Alexander polynomial Δ exactly. Writing [p]_t·Δ up to a unit as Δ̃(u) = Σ_{m=0}^{r−1} γ_m(t) u^m with u = t^{rs}, we prove (for 0 < q < p; for q > p sketched in a remark and checked by machine for p ≤ 10) a multiset law expressing each γ_m·[r]_t as a sum of p monomials with all coefficients +1, a closed formula for the degree sequence d_m = top γ_m (highest exponent) in terms of return times of a rotation, and the resulting leading-coefficient law |lc(Δ)| = #{m : d_m + mrs maximal}.

Theta on constructed Conway-mutant pairs: thirty separated, one not

2026-10-22T00:00:00Z

Thirty pairs of knots related by a Conway mutation, each pair given with both planar diagrams and with the value of the Bar-Natan/van der Veen Theta invariant on both members. NO THEOREM IS CLAIMED AND NONE OF THE MATHEMATICS IS OURS: Theta is theirs (arXiv:2509.18456, arXiv:2109.02057), mutation is classical, and we do not know what is already published about Theta and mutation. What is deposited is a set of computed cases, built because the published supply of mutant test cases for this invariant is small. Construction: each pair is a pretzel knot P(2,b,c,d) against P(2,c,b,d), the two middl

Lower bounds for the unknotting number over a corpus of built curves

2026-10-22T00:00:00Z

VERSION 3 (17 October 2026) - STATUS UPDATE, READ FIRST: measured against B. A. Burton's census (up to 19 crossings) and Maguire's census (up to 17) and the corrected novelty boundary (all prime knots enumerated up to 20 crossings, alternating ones up to 24), 8 of the 13 knots with u determined are table knots (calibration, not a result), 4 have a proven crossing number of only c >= 20 (inside the enumerated range), and only 1 lies beyond every existing enumeration (crossing number 24, proven, non-alternating, u = 3). The '5 beyond any enumeration' below counted crossings of our diagrams, not

Congruences on the Conway polynomial forced by Conant's Corollary 7.3: an answer to the first half of Question 9.2

2026-10-22T00:00:00Z

VERSION 2 (17 October 2026). An answer to the modulo-8 part of the first half of Question 9.2 of J. Conant, 'A proof of the mod 4 Kawauchi conjecture', arXiv:2607.18655 (21 July 2026), which asks which congruences modulo 8 and higher powers of 2 on the Conway polynomial follow from the presentation eps_l [l] Nabla_K = M^2 - 4 E^2 of his Corollary 7.3, beyond the constant-term constraint E(0) = (l^2-1)/8 mod 2. Writing w = z^2 and N = eps_l [l] Nabla_K, the substitution U = M - 2E, V = M + 2E turns the presentation into a factorisation N = U V with V congruent to U modulo 4 and U(0) odd, hence

Priority deposit v5: 338 positive amphicheiral hyperbolic knots whose Conway polynomial does not split -- Part III, order 16 and a discrepancy in the rotation parameter

2026-10-24T00:00:00Z

Version 5 (prepared 23 October 2026): corrections to the text of version 4 (compiled 19 October 2026, on Zenodo 22 October 2026); data and existing scripts are unchanged, and no number of version 4 changes. The main correction is CORRECTION IN VERSION 5 below; the others are the sign convention of l, the rotation parameter (it depends on the choice of generator), the scope of the cited results of Conant, and the honesty note; the full list is in README.md. Version 4 superseded version 3 (315 knots) on the same concept DOI; the 315 records of version 3 are carried over unchanged in every value.

Priority deposit: c7 inequality at 14/15 crossings, Knot 15331 bounds, and a blind Farey/precessing-ellipse knot atlas

2026-10-22T00:00:00Z

Three dated computational results, deposited to fix priority in permanent, non-editable, citable form. (1) The inequality spP_a <= spF_a (HOMFLY a-spread <= Kauffman a-spread; candidate (c11)/(c10)/(c7) of versions 1/2/3-4 of M. Jablonowski, arXiv:2605.22652, proved there for alternating knots and screened up to 13 crossings; version 5 no longer lists it) verified beyond 13 crossings for the first time. VERSION 2 (19 October 2026): the July tables (46,972 knots at c=14; 253,281 of the 253,293 knots at c=15) were computed without a validity check on the Kauffman polynomial, and the tool used (k