Trisections with Kirby-Thompson length 2

TL;DR

This paper proves Kirby and Thompson's conjecture that a 4-manifold with a trisection of Kirby-Thompson length 2 must have length 0, clarifying the relationship between these length measures.

Contribution

It establishes that trisections with Kirby-Thompson length 2 necessarily correspond to 4-manifolds with length 0, confirming a conjecture by Kirby and Thompson.

Findings

Length 2 trisections imply length 0 for the 4-manifold

Confirmed Kirby and Thompson's conjecture

Clarified the relationship between trisection lengths

Abstract

Kirby and Thompson introduced a length of a trisection. They also defined the length of a 4-manifold as the minimum of length among all lengths of trisection of a 4-manifold. In this paper, we consider trisections whose Kirby-Thompson length is 2. Kirby and Thompson conjectured that length 2 trisection is a trisection of 4-manifold with length 0. We shall prove this conjecture in this paper.