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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Bridge trisections of knotted surfaces in $S^4$
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by Jeffrey Meier and Alexander Zupan PDF
Trans. Amer. Math. Soc. 369 (2017), 7343-7386 Request permission

Abstract:

We introduce bridge trisections of knotted surfaces in the 4–sphere. This description is inspired by the work of Gay and Kirby on trisections of 4–manifolds and extends the classical concept of bridge splittings of links in the 3–sphere to four dimensions. We prove that every knotted surface in the 4–sphere admits a bridge trisection (a decomposition into three simple pieces) and that any two bridge trisections for a fixed surface are related by a sequence of stabilizations and destabilizations. We also introduce a corresponding diagrammatic representation of knotted surfaces and describe a set of moves that suffice to pass between two diagrams for the same surface. Using these decompositions, we define a new complexity measure: the bridge number of a knotted surface. In addition, we classify bridge trisections with low complexity, we relate bridge trisections to the fundamental groups of knotted surface complements, and we prove that there exist knotted surfaces with arbitrarily large bridge number.
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Additional Information
  • Jeffrey Meier
  • Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47408
  • MR Author ID: 849257
  • Email: jlmeier@indiana.edu
  • Alexander Zupan
  • Affiliation: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588
  • MR Author ID: 863648
  • Email: zupan@unl.edu
  • Received by editor(s): August 17, 2015
  • Received by editor(s) in revised form: March 7, 2016
  • Published electronically: May 30, 2017
  • Additional Notes: The first author was supported by NSF grant DMS-1400543
    The second author was supported by NSF grant DMS-1203988
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 7343-7386
  • MSC (2010): Primary 57M25, 57Q45
  • DOI: https://doi.org/10.1090/tran/6934
  • MathSciNet review: 3683111