Bounding the Kirby-Thompson invariant of spun knots

TL;DR

This paper establishes initial bounds for the Kirby-Thompson invariant of spun knots, including a specific value of 15 for the spun trefoil, advancing understanding of their complexity.

Contribution

It provides the first significant bounds for the Kirby-Thompson invariant of spun knots, including exact calculations for the spun trefoil.

Findings

Kirby-Thompson invariant of spun trefoil is 15

First significant bounds for this invariant of spun knots

Bridge trisection complexity relates to pants complex distances

Abstract

A bridge trisection of a smooth surface in $S^4$ is a decomposition analogous to a bridge splitting of a link in $S^3$. The Kirby-Thompson invariant of a bridge trisection measures its complexity in terms of distances between disc sets in the pants complex of the trisection surface. We give the first significant bounds for the Kirby-Thompson invariant of spun knots. In particular, we show that the Kirby-Thompson invariant of the spun trefoil is 15.