Paper · Preprint · version 2 of 2026-10-22, first deposited 2026-10-03
Werner Alois Stanggassinger — in collaboration with Claude Code (Anthropic)
Permanent archive (all files, all versions): doi:10.5281/zenodo.23117903 (Zenodo, CERN).
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 plateau is carried by one chain of Hartley's move T1. The proof is an induction over Hartley's moves using his bottom sequences, as presented by W. Chen (arXiv:1712.04993). As a by-product the induction gives a written-out proof of Chen's theorem that the Hirasawa-Murasugi bound holds for two-bridge knots and links (in the form Z <= |sigma|); in Chen's proof (arXiv and journal versions) the induction step for T1 leaves the case of even length (from a link to a knot) as 'similar'. WHAT IS NOT OURS: the description of the Alexander coefficients as occupation numbers of a walk is Hartley's (1979), Minkus' (1982) and Hoste's (2020); Hartley's identity and IH properties are used as stated in Chen's paper; the Hirasawa-Murasugi bound for two-bridge knots is Chen's theorem; when the plateau is carried by the top T1-chain, the bound is obtained as in Chen's proof of his Prop. 3.12 and the drop of Z and sigma by two follows from his Prop. 3.2, Prop. 3.4 and Lemma 3.9 (Chen does not state this descent); that T1 raises the crossing number by one is Prop. 8(a) in Section 6.3 of R. Gaebler's senior thesis (Harvey Mudd College, 2004). What is new, as far as we could find, is the descent when the plateau is carried by a deeper T1-chain, which does not reduce to the top chain by Schubert's symmetry (example N(39/14), continued fraction [2,1,3,1,2], up to mirror image the only plateau knot with at most nine crossings of which no representative p/q' has the plateau on its top chain), the reading of a T1-chain with at least two moves as a coherently oriented twist region, and, for a deeper chain, c(K') = c(K) - 2. STATUS: not refereed; how the work was done and checked is stated in the section on tool use at the end of the paper. The main lemmas and both theorems were also tested by machine on finite ranges (testing, not formal verification): code.zip contains a pure-Python script that recomputes every count in Section 8 of the paper except the signature comparison (needs SageMath), the separate check of Lemma 2, the carrying-region item and the two re-implementations (their own pure-Python scripts are included), the rest of the last item and the census remark (its scripts are included to document the method; its census inputs are not in the deposit), in about two minutes; the README maps every item to its script and output file (expected outputs in data.zip). A remark reports census data beyond two-bridge knots (all 20158 hyperbolic alternating prime knots up to 19 crossings with plateau half-width 1, counted up to mirror image, admit such a crossing change; B. A. Burton's census). SUPPLEMENT (not part of the paper, data/ajk_cor316 in data.zip): the conclusions of Corollary 3.16 of Azarpendar, Juhasz and Kalman hold for all 2120 H-M sharp alternating prime knots with a plateau up to 19 crossings (counted up to mirror image), in the census diagram (1996 within the hypothesis of the corollary, 124 outside it); the summand K_1 is a torus link T(2,q) in every case; their Figure 56 shows 14 of the 16 sharp prime knots with a plateau up to 10 crossings (8_4 and 10_4 are not in it). This work was done by Werner Alois Stanggassinger in collaboration with Claude Code (Anthropic): the direction of the work and the hardware are Stanggassinger's; the proofs were worked out by Claude Code; the write-up lists Stanggassinger as author and states this in a footnote and in a section on tool use. VERSION 2 (17 October 2026), after an outside reading: author name written in full, date line without a status sentence, new section on tool use, the deposit's DOI and Chen's journal version (Prop. 3.9, Case (2)) cited, and the scripts replaced by English versions of the working scripts (only names, comments and printed text translated; see code/NAME_MAP.md), re-run on 17 October 2026 with outputs equal, number by number, to those of version 1. The mathematics is unchanged. Licences: text CC BY 4.0; code GNU General Public License v3.0 or later; data CC BY 4.0.
Keywords: knot theory · two-bridge knots · Alexander polynomial · Fox trapezoidal conjecture · Hirasawa-Murasugi conjecture · signature · twist region · Hartley · alternating knots
@misc{stanggassinger2026alexander,
author = {Stanggassinger, Werner Alois},
title = {The Alexander plateau of a two-bridge knot is carried by one coherent twist region},
year = {2026},
publisher = {Zenodo},
doi = {10.5281/zenodo.23117903},
url = {https://knot-structures.stainlesssteel4u.de/preprints/alexander-plateau-two-bridge-knots/}
}This page mirrors the Zenodo record; if the two ever differ, the Zenodo record is authoritative.