Essential-Surface Complexes of Knot Exteriors: Image, Kernel, and Reconstruction

[2607.20869] Essential-Surface Complexes of Knot Exteriors: Image, Kernel, and Reconstruction Skip to main content Search Submit Donate Log in Mathematics > Geometric Topology arXiv:2607.20869 (math) [Submitted on 23 Jul 2026 ( v1 ), last revised 27 Oct 2026 (this version, v3)] Title: Essential-Surface Complexes of Knot Exteriors: Image, Kernel, and Reconstruction Authors: Makoto Ozawa View PDF HTML (experimental) Abstract: This is an expository article. The curve complex of a surface is deliberately forgetful, yet in most cases its automorphism group recovers the mapping class group. We explain what happens when the same question is asked for the essential-surface complex $\mathcal{ES}(E)$ of a knot exterior $E=E(K)$, equivalently Schultens's initial surface complex $\mathcal{S}_0(E)$. The discussion is organized around three questions: which combinatorial symmetries are geometric (the image), which homeomorphisms are invisible (the kernel), and how much of $E$ can be read off from the complex (reconstruction). The basic objects are first drawn in classical examples, and the general framework is introduced only afterwards. We collect the elementary structural facts that organize the subject -- the flag property, the layering by boundary slope, finite dimension, unbounded dimension for iterated cables, and Kakimizu's complexes as full subcomplexes -- and compare with handlebodies and connected sums of $S^2\times S^1$, where image and kernel are known completely. Along the way we record two short consequences that we have not found stated in this form: for a slope-separated knot every orientation-preserving mapping class acts trivially, and the image is $\mathbb{Z}/2$ exactly when the knot is amphichiral and a nonzero boundary slope occurs; and for a connected sum of nonfibered prime knots the decomposing-annulus twist translates Banks's winding coordinate. A table separates what is classical from what is recorded here. The article closes with a three-task formulation of the rigidity problem, a list of problems with their current status, a worked two-bridge recipe, and a guide to further reading. Comments: 41 pages, 17 figures, 1 table. v3: rewritten as an expository article; added a status table of classical vs. new results, a comparison with handlebodies and #S^2xS^1, a problem list with status, and further references; corrected the two-bridge continued-fraction convention Subjects: Geometric Topology (math.GT) MSC classes: 57K10 (Primary) 57-02, 57K30, 57M60 (Secondary) Cite as: arXiv:2607.20869 [math.GT] (or arXiv:2607.20869v3 [math.GT] for this version) https://doi.org/10.48550/arXiv.2607.20869 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Makoto Ozawa [ view email ] [v1] Thu, 23 Jul 2026 02:49:29 UTC (43 KB) [v2] Sun, 23 Aug 2026 01:34:26 UTC (49 KB) [v3] Tue, 27 Oct 2026 06:17:36 UTC (54 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.GT < prev | next > new | recent | 2026-07 Change to browse by: math 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)

Related documents

• Essential-Surface Complexes of Knot Exteriors: Image, Kernel, and Reconstruction#665872
• Knot Floer homology of boundary Dehn twists#1070596
• Simon's knot genus problem and Lewin $3$-manifold groups#987625
• Skein relations for invariants of Legendrian surfaces#617071
• Skein relations for invariants of Legendrian surfaces#616571