Bridge trisections and classical knotted surface theory

Abstract

We seek to connect ideas in the theory of bridge trisections with other well-studied facets of classical knotted surface theory. First, we show how the normal Euler number can be computed from a tri-plane diagram, and we use this to give a trisection-theoretic proof of the Whitney-Massey Theorem, which bounds the possible values of this number in terms of the Euler characteristic. Second, we describe in detail how to compute the fundamental group and related invariants from a tri-plane diagram, and we use this, together with an analysis of bridge trisections of ribbon surfaces, to produce an infinite family of knotted spheres that admit non-isotopic bridge trisections of minimal complexity. Third, we adapt Seifert's algorithm to the setting of tri-plane diagrams, which allows for the construction of a Seifert solid that is described by a Heegaard diagram. Finally, we give a classification result that says that a b-bridge trisection in which some sector contains at least b − 1 patches is completely decomposable, and thus the corresponding surface is unknotted. Corollary 3.9. Let D = (D 1 , D 2 , D 3 ) be a tri-plane diagram of a surface S ⊂ S 4 . Let w i be the writhe of the diagram D i ∪ D i+1 . Then e(S) = w 1 + w 2 + w 3 .As one application, we obtain the following well-known result.