Some lower bounds for the Kirby-Thompson invariant

TL;DR

This paper establishes lower bounds for the Kirby-Thompson invariant of certain 4-manifolds, computes it for the spin of L(2,1), and demonstrates the existence of 4-manifolds with arbitrarily large invariants.

Contribution

It provides the first example of a 4-manifold with a non-trivial Kirby-Thompson invariant and shows that these invariants can be arbitrarily large.

Findings

Lower bounds for the Kirby-Thompson invariant of specific 4-manifolds.

Determination of the invariant for the spin of L(2,1).

Existence of 4-manifolds with arbitrarily large Kirby-Thompson invariant.

Abstract

Kirby and Thompson introduced a non-negative integer-valued invariant, called the Kirby-Thompson invariant, of a $4$-manifold using trisections. In this paper, we give some lower bounds for the Kirby-Thompson invariant of certain $4$-manifolds. As an application, we determine the Kirby-Thompson invariant of the spin of $L(2,1)$, which is the first example of a $4$-manifold with non-trivial Kirby-Thompson invariant. We also show that there exist $4$-manifolds with arbitrarily large Kirby-Thompson invariant.