Vol. 319, No. 2, 2022

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Bridge trisections and classical knotted surface theory

Jason Joseph, Jeffrey Meier, Maggie Miller and Alexander Zupan

Vol. 319 (2022), No. 2, 343–369
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 nonisotopic bridge trisections of minimal complexity.

Keywords
knotted surface, bridge trisection, 2-knot, trisection
Mathematical Subject Classification
Primary: 57K45
Milestones
Received: 30 December 2021
Revised: 20 May 2022
Accepted: 27 May 2022
Published: 11 September 2022
Authors
Jason Joseph
Department of Mathematics
Rice University
Houston, TX
United States
Jeffrey Meier
Department of Mathematics
Western Washington University
Bellingham, WA
United States
Maggie Miller
Department of Mathematics
Stanford University
Stanford, CA
United States
Alexander Zupan
Department of Mathematics
University of Nebraska–Lincoln
Lincoln, NE
United States