Paper · Preprint · version 8 of 2026-11-06, first deposited 2026-09-15
Werner Alois Stanggassinger — in collaboration with Claude Code (Anthropic)
Permanent archive (all files, all versions): doi:10.5281/zenodo.22769534 (Zenodo, CERN).
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}. Consequently Δ is monic if and only if that maximum is attained once, and Δ fails to be monic exactly when −rs is an edge slope of the upper convex hull of {(m, d_m)}; in particular at most r−1 values of s are non-monic for fixed (p,q,r), however large |s|. We then classify completely, for every admissible parameter, when the r points (m, d_m) are collinear: for 0 < q < p by three explicit conditions on the first-return map of a rotation (rendered by a proposition, proved here, as one arithmetic condition on (p,q,r) for each), for q > p by the dual criterion (proved for the orbit sequence M_j; that it describes d_m for q > p rests on a sketch, checked by machine for p ≤ 10), and for q < 0 through the mirror. The extremal case |lc(Δ)| = r is exactly the collinear case whose common difference st is divisible by r, at the single twist s = −st/r (for q > 0). This gives an exact algebraic criterion for the monic side of a conjecture of Adnan and Park; we do not address its geometric direction.
Prior work. Two papers by Sangyop Lee (J. Knot Theory Ramifications 34 (2025) 2450051, and J. Knot Theory Ramifications (2026) 2650033) concern the same family at r = 2. The first determines, for a negative full twist (s = −1), which of these knots are cable or torus knots; the second shows that every such knot outside an explicit list of unknots, torus knots and cables is hyperbolic. On that list our leading-coefficient law agrees with the Alexander polynomial (327 cases checked); for the hyperbolic knots the classification does not give the Alexander polynomial, so the case r = 2 of our law does not follow from it. For r ≥ 3 we are not aware of a classification from which our statements would follow. The case q | r with q ≥ 2 (Δ always monic) follows from Theorem 1 of Lee (J. Knot Theory Ramifications 21 (2012) 1250005).
Version 8 (5 November 2026) proves the proposition that renders the collinearity criteria for 0 < q < p as arithmetic in (p,q,r), which up to version 7 was stated with the mechanism of its proof and checked by machine, and sharpens it: (K2) holds if and only if rμ < q, and (K1) if and only if rν < q, where qq̄ = 1 + pμ with q̄ the inverse of q modulo p and ν = q − μ; the four-condition form of the earlier versions remains correct. Part of the proof is due to DeepSeek (in answer to our question, read and repaired by us), the rest is ours; two lemmas are added, and a script (with a small library) and its output, which check the statements and the main intermediate steps on all 193,956 triples with p ≤ 100. Version 7 (4 November 2026) proves the identity between our multiset law and Bryant's formula: an explicit bijection (Proposition in the Introduction, found by DeepSeek in answer to our question, verified by hand and by machine) shows that each block of his double sum is γ_m·[r]_t exactly, so the two numerators coincide as polynomials; the wording of the comparison is sharpened accordingly, and one script with its output is added. Version 6 (4 November 2026) adds a citation of Bryant, arXiv:2611.03265v1, who computes Δ, its degree and its leading coefficient by Dehn surgery for 1 < q < p, 1 ≤ r ≤ p−1 and all s; grouped by powers of u, his formula equals our multiset law on all 40,317 triples with p ≤ 60 that we checked (one new script; up to a common power of t this follows from the two theorems, the check shows that no extra power occurs), and nothing from it is used. Version 5 (3 November 2026) adds a citation of Adnan and Park, arXiv:2610.34813v1, who treat the disjoint range max(|p|,|q|) < r ≤ |p|+|q|; nothing from it is used, and the scripts and outputs are unchanged from version 4. Version 4 (17 October 2026; revised 19, 23, 24 and 25 October 2026) deposits, besides the note (PDF and LaTeX source), the scripts and outputs behind its Section "Verification" and the verification paragraphs of Section 6; versions 1 to 3 deposited the PDF only. Among other corrections, the revision of 23 October corrects three statements that version 3 also contained: the decomposition above is of [p]_t·Δ, not of Δ itself; the multiset law concerns γ_m·[r]_t, not γ_m; and the extremal case |lc(Δ)| = r needs, beyond collinearity, that r divides st (version 3 equated the two). The leading-coefficient law, the convex-hull criterion and the classification of collinearity are unaffected. The proof of the multiset law is given in the note as an outline; it is written out in the companion note for 0 < q < p, and the case q > p is sketched in a remark and checked by machine for p ≤ 10. The changes against version 3 are listed in README.md (version history). Licences: text CC BY 4.0; code GNU General Public License v3.0 or later; data CC BY 4.0.
This work was developed in collaboration with the AI system Claude Code (Anthropic): the choice of problem, the direction of the work and the computing hardware are due to the author (a sculptor); the formalisation, the algorithms and the proofs were developed by Claude Code (Anthropic) working with the author; the bijection of the Proposition in the Introduction (version 7) and part of the proof of the arithmetic form of the collinearity criteria (version 8) are due to DeepSeek, as the note's section on tool use states. The collaboration is stated as a fact, not a disclaimer; details are in the section on tool use at the end of the note. The author takes full responsibility for all contents.
Website: https://knot-structures.stainlesssteel4u.de/
Keywords: twisted torus knots · Alexander polynomial · leading coefficient · fibered knots · monic Alexander polynomial · convex hull · computer-assisted proof · 57K10
@misc{stanggassinger2026leading,
author = {Stanggassinger, Werner Alois},
title = {The leading coefficient of the Alexander polynomial of twisted torus knots},
year = {2026},
publisher = {Zenodo},
doi = {10.5281/zenodo.22769534},
url = {https://knot-structures.stainlesssteel4u.de/preprints/leading-coefficient-twisted-torus-knots/}
}This page mirrors the Zenodo record; if the two ever differ, the Zenodo record is authoritative.