Toric mosaic numbers of eighteen ten-crossing knots
Year: 2026
Type: preprint
Abstract: Heiney, Kipe, Pezzimenti, Pontes and Ta (arXiv:2504.02265v1; Topology Appl. 377 (2026) 109657, doi:10.1016/j.topol.2025.109657) studied the toric mosaic number mT(K) and tabulated it for all knots with at most ten crossings, except for eighteen knots for which their Table 4 gives only 'mT >= 4, n/a'. From a complete census of the 4,934,695,175 suitably connected toric 4-mosaics and a random search over toric 5-mosaics we find mT(10_129) = 4 and mT = 5 for the other seventeen: 10_17, 10_52, 10_76, 10_81, 10_86, 10_88, 10_90, 10_96, 10_97, 10_98, 10_105, 10_110, 10_114, 10_115, 10_119, 10_120 and 10_121 (KnotInfo names). The complete censuses of toric 2- and 3-mosaics also show that three entries of the arXiv version v1 of the source are incorrect or mutually inconsistent, two of them also in the published version: 10_145 is not a toric 3-mosaic knot, mT(5_2) = 3 (corrected in the published version), and mT(10_132) >= 4, with equality by a toric 4-mosaic. METHOD: every toric 4-mosaic passing an invariant filter that cannot reject a diagram of a target is traced with the authors' program and separated by the exact Jones polynomial; the only toric 4-mosaics carrying the Jones polynomial of one of the seventeen knots or its mirror image are 256 with that of 10_119, and they represent 12n_431. For every 5-mosaic listed in the note the complement of its diagram is isometric to the complement of the KnotInfo diagram of the knot; both chiralities were found for the thirteen chiral knots. The result is computer-assisted: it rests on the agreement of our trace with the authors' code, the completeness of the enumeration (checked against the counts of Oh, Hong, Lee, Lee and Yeon), the filter, the GPU kernel (its hits agree with a CPU recomputation of the whole block in both runs), the exact Jones polynomial and the identifications. CONTENTS: the note (PDF and LaTeX source, dated 19 October 2026, revised 11 November 2026); the hit lists of both 4-mosaic runs; the separation by Jones polynomial; the identifications and witnesses; the CPU recomputations of the GPU blocks; the 10_132 census; the comparisons with the authors' code; the invariant checks used in the proof; and the programs, except three files that contain our port of the authors' trace (withheld because the authors' code carries no licence; checksums in the README). README.md maps every statement of the note to the file that checks it. The authors' code (github.com/margekk/toric-mosaics-2025, commit 878037941c3a) is not included. HONESTY NOTES: no invariant is computed by an AI, the computations being done by Python and C programs using SnapPy 3.3.2, Regina 7.4, SageMath 10.9, database_knotinfo 2026.10.5 and CuPy 14.2.0. 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. Details are in the section on tool use at the end of the note. STATUS: not refereed; the arguments and computations have not been checked by a professional mathematician. Licences: text CC BY 4.0; code GNU General Public License v3.0 or later; data CC BY 4.0.
Author: Stanggassinger, Werner Alois
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225798713