# State graphs and fibered state surfaces

## TL;DR

This paper introduces an algebraic criterion based on state graphs to determine when checkerboard surfaces of knots and links are fibers, enabling classification of fibering properties for various families, including 2-bridge links.

## Contribution

It provides a new algebraic condition to identify fibered checkerboard surfaces directly from state graphs, advancing the understanding of fibering in knot theory.

## Key findings

- Algebraic condition characterizes fibered checkerboard surfaces.
- Classifies fibering for families of planar graphs.
- Determines fibering for 2-bridge links.

## Abstract

Associated to every state surface for a knot or link is a state graph, which embeds as a spine of the state surface. A state graph can be decomposed along cut-vertices into graphs with induced planar embeddings. Associated with each such planar graph is a checkerboard surface, and each state surface is a fiber if and only if all of its associated checkerboard surfaces are fibers. We give an algebraic condition that characterizes which checkerboard surfaces are fibers directly from their state graphs. We use this to classify fibering of checkerboard surfaces for several families of planar graphs, including those associated with 2-bridge links. This characterizes fibering for many families of state surfaces.