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Bridge trisections of knotted surfaces in $ S^4$


Authors: Jeffrey Meier and Alexander Zupan
Journal: Trans. Amer. Math. Soc. 369 (2017), 7343-7386
MSC (2010): Primary 57M25, 57Q45
DOI: https://doi.org/10.1090/tran/6934
Published electronically: May 30, 2017
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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
Email: jlmeier@indiana.edu

Alexander Zupan
Affiliation: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588
Email: zupan@unl.edu

DOI: https://doi.org/10.1090/tran/6934
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
Article copyright: © Copyright 2017 American Mathematical Society