Bridge trisections and classical knotted surface theory

TL;DR

This paper connects bridge trisections with classical knotted surface theory, providing new methods to compute invariants and demonstrating the existence of knotted spheres with multiple minimal bridge trisections.

Contribution

It introduces a way to compute the normal Euler number and fundamental group from tri-plane diagrams, linking bridge trisections to classical invariants and showing multiple minimal bridge trisections for certain knotted spheres.

Findings

Normal Euler number can be computed from tri-plane diagrams.

A new proof of the Whitney-Massey Theorem using trisections.

Existence of knotted spheres with non-isotopic minimal bridge trisections.

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.