Minimal genus relative trisections of corks

TL;DR

This paper determines the minimal genus of relative trisections for corks, specifically proving the Akbulut cork has genus 3, and constructs examples of corks and exotic pairs with low genus, advancing understanding of 4-manifold decompositions.

Contribution

It establishes the trisection genus of the Akbulut cork as 3 and provides new examples of corks and exotic pairs with minimal genus diagrams.

Findings

The trisection genus of the Akbulut cork is 3.

Constructed infinitely many corks with trisection genus 3.

Provided low genus relative trisection diagrams for exotic 4-manifolds.

Abstract

In this paper, we prove that the trisection genus of the Akbulut cork is $3$ and construct infinitely many corks with trisection genus $3$. These results give the first examples of contractible $4$-manifolds whose trisection genera are determined except for the $4$-ball. We also give a lower bound for the trisection genus of a $4$-manifold with boundary. In addition, we construct low genus relative trisection diagrams of an exotic pair of simply-connected $4$-manifolds with $b_2 = 1$.