Fiber detection for state surfaces

TL;DR

This paper characterizes when a state surface derived from a link diagram is a fiber surface, linking it to the graph structure and Jones polynomial coefficients, simplifying previous proofs in knot theory.

Contribution

It provides a new criterion for fibered links based on the associated graph being a tree, and relates Jones polynomial coefficients to fiber surface obstructions.

Findings

A homogeneous state surface is a fiber if and only if its associated graph is a tree.

Second and next-to-last Jones polynomial coefficients obstruct certain state surfaces from being fibers.

Simplifies previous proofs regarding fibered links and state surfaces.

Abstract

Every Kauffman state \sigma of a link diagram D(K) naturally defines a state surface S_\sigma whose boundary is K. For a homogeneous state \sigma, we show that K is a fibered link with fiber surface S_\sigma if and only if an associated graph G'_\sigma is a tree. As a corollary, it follows that for an adequate knot or link, the second and next-to-last coefficients of the Jones polynomial are obstructions to certain state surfaces being fibers for K. This provides a dramatically simpler proof of a theorem from [arXiv:1108.3370].