[2009.13461] Embedded surfaces with infinite cyclic knot group Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Geometric Topology arXiv:2009.13461 (math) [Submitted on 28 Sep 2020 ( v1 ), last revised 1 May 2026 (this version, v7)] Title: Embedded surfaces with infinite cyclic knot group Authors: Anthony Conway , Mark Powell View a PDF of the paper titled Embedded surfaces with infinite cyclic knot group, by Anthony Conway and Mark Powell View PDF HTML (experimental) Abstract: We study locally flat, compact, oriented surfaces in $4$-manifolds whose exteriors have infinite cyclic fundamental group. We give algebraic topological criteria for two such surfaces, with the same genus $g$, to be related by an ambient homeomorphism, and further criteria that imply they are ambiently isotopic. Along the way, we prove that certain pairs of topological $4$-manifolds with infinite cyclic fundamental group, homeomorphic boundaries, and equivalent equivariant intersection forms, are homeomorphic. Comments: v2 fixes an error in the proof of Theorem 1.3. The issue in the proof Theorem 5.10 (now Theorem 5.11) has been corrected. v3, v4 are reorganisations; new figures and applications are added. v5: Added report number. v6: Fixed the definition of a trivial 1-handle stabilisation. To appear in Geometry & Topology. v7: Fixes an error: Theorems 1.7, 1.8 on n-roll 1-twist rim surgery only hold for n=0 Subjects: Geometric Topology (math.GT) MSC classes: 57K40, 57K10, 57N35, Report number: MPIM-Bonn-2021 Cite as: arXiv:2009.13461 [math.GT] (or arXiv:2009.13461v7 [math.GT] for this version) https://doi.org/10.48550/arXiv.2009.13461 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Geom. Topol. 27 (2023) 739-821 Related DOI : https://doi.org/10.2140/gt.2023.27.739 Focus to learn more DOI(s) linking to related resources Submission history From: Anthony Conway [ view email ] [v1] Mon, 28 Sep 2020 16:45:21 UTC (84 KB) [v2] Fri, 9 Oct 2020 19:36:20 UTC (84 KB) [v3] Wed, 11 Nov 2020 16:58:49 UTC (85 KB) [v4] Thu, 10 Jun 2021 06:07:36 UTC (424 KB) [v5] Tue, 19 Oct 2021 22:58:54 UTC (424 KB) [v6] Sun, 28 Nov 2021 16:46:31 UTC (424 KB) [v7] Fri, 1 May 2026 16:01:33 UTC (424 KB) Full-text links: Access Paper: View a PDF of the paper titled Embedded surfaces with infinite cyclic knot group, by Anthony Conway and Mark Powell View PDF HTML (experimental) TeX Source view license Current browse context: math.GT < prev | next > new | recent | 2020-09 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: 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) Major funding support from