Bounds for Kirby-Thompson invariants of knotted surfaces

TL;DR

This paper establishes precise lower bounds for two variants of Kirby-Thompson invariants in knotted surfaces, including a new version based on the dual curve complex, and computes their exact values for many surfaces.

Contribution

It introduces a new version of the Kirby-Thompson invariant based on the dual curve complex and provides exact calculations for numerous knotted surfaces.

Findings

Sharp lower bounds for two Kirby-Thompson invariants.

Exact values computed for infinitely many knotted surfaces.

Introduction of a dual curve complex-based invariant.

Abstract

We provide sharp lower bounds for two versions of the Kirby-Thompson invariants for knotted surfaces, one of which was originally defined by Blair, Campisi, Taylor, and Tomova. The second version introduced in this paper measures distances in the dual curve complex instead of the pants complex. We compute the exact values of both KT-invariants for infinitely many knotted surfaces with bridge number at most six.