Fiber detection for state surfaces

Abstract

Every Kauffman state $\sigma$ of a link diagram $D\left(K\right)$ 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 ${\mathbb{G}}_{\sigma }^{\prime }$ 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 the obstructions to certain state surfaces being fibers for $K$.

This provides a dramatically simpler proof of a theorem of the author with Kalfagianni and Purcell.