Kirby-Thompson invariants of distant sums of standard surfaces

TL;DR

This paper computes the Kirby-Thompson invariants for finite distant sums of standard surfaces, providing the first examples where these invariants and bridge numbers can be arbitrarily large, advancing understanding of knotted surface complexity.

Contribution

It determines the L and L* invariants for any finite distant sum of standard surfaces, introducing new examples with unbounded invariants and bridge numbers.

Findings

L and L* invariants can be arbitrarily large for certain knotted surfaces.

First explicit examples of knotted surfaces with unbounded bridge numbers and invariants.

Provides formulas or methods to compute invariants for sums of standard surfaces.

Abstract

Blair, Campisi, Taylor, and Tomova defined the L-invariant L(F) of a knotted surface F, using pants complexes of trisection surfaces of bridge trisections of F. After that, Aranda, Pongtanapaisan, and Zhang introduced the L*-invariant L*(F) using dual curve complexes instead of pants complexes. In this paper, we determine both of L-invariant and L*-invariant of any finite distant sum of standard surfaces, and this is the first example of knotted surfaces whose bridge numbers and these invariants can be arbitrary large.