Bridge trisections of knotted surfaces in 𝑆⁴

Meier, Jeffrey; Zupan, Alexander

doi:10.1090/tran/6934

Bridge trisections of knotted surfaces in $S^4$
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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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