Maps on Surfaces and the Tabulation of Knots and Links in the Thickened Torus Through Ten Crossings

[2601.15512] Maps on Surfaces and the Tabulation of Knots and Links in the Thickened Torus Through Ten Crossings Skip to main content Search Submit Donate Log in Mathematics > Combinatorics arXiv:2601.15512 (math) [Submitted on 21 Jan 2026 ( v1 ), last revised 27 Aug 2026 (this version, v2)] Title: Maps on Surfaces and the Tabulation of Knots and Links in the Thickened Torus Through Ten Crossings Authors: Alexander Omelchenko View PDF HTML (experimental) Abstract: We give a combinatorial tabulation of knots and links in the thickened torus $T^2 \times I$ based on the theory of maps on surfaces: cellular $4$-regular torus projections are encoded by permutation pairs, and unsensed equivalence classes are enumerated completely and without duplication by canonical representatives. The method is validated against published genus-one tables for $N \le 5$ and extended to $N=6,7,8,9,10$, producing, to our knowledge, the first complete tabulation of prime knot and link diagram types in the fixed thickened torus $T^2 \times I$ at these crossing numbers, under the conventions stated below -- more than $1.3$ million knot and link types in total. The computation yields three proved structural results on straight-ahead components, bigon faces, and the parity of the $a$-span of the genus-one bracket, together with several conjectures, including a $4N$ bound on the $a$-span in the spirit of the Kauffman--Murasugi--Thistlethwaite theorem. A reference implementation and machine-readable datasets are provided. Comments: 39 pages. Substantially expanded and retitled version; the tabulation is extended from eight to ten crossings. Reference implementation and machine-readable datasets are provided Subjects: Combinatorics (math.CO) ; Geometric Topology (math.GT) MSC classes: 57K10, 05C30, 05C10, 57K12 Cite as: arXiv:2601.15512 [math.CO] (or arXiv:2601.15512v2 [math.CO] for this version) https://doi.org/10.48550/arXiv.2601.15512 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Alexander Omelchenko [ view email ] [v1] Wed, 21 Jan 2026 22:48:17 UTC (31 KB) [v2] Thu, 27 Aug 2026 22:44:31 UTC (49 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2026-01 Change to browse by: math math.GT References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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