# Critical Bridge Spheres for Links with Arbitrarily Many Bridges

## TL;DR

This paper constructs infinite families of links with arbitrarily many bridges that admit a special type of critical bridge sphere, expanding understanding of bridge sphere properties in link theory.

## Contribution

It introduces the first examples of links with any number of bridges that have critical bridge spheres, demonstrating their existence for all integers b ≥ 3.

## Key findings

- Existence of links with arbitrarily many bridges having critical bridge spheres
- Construction of infinite families of such links for each b ≥ 3
- Introduction of 'square' links with these properties

## Abstract

We show that for every integer $b\geq 3$, there exists a link in a $b$-bridge position with respect to a critical bridge sphere. In fact, for each $b$, we construct an infinite family of links which we call square whose bridge spheres are critical.