#### Volume 13, issue 5 (2013)

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Fiber detection for state surfaces

### David Futer

Algebraic & Geometric Topology 13 (2013) 2799–2807
##### 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.

##### Keywords
adequate knot, homogeneous knot, spanning surface, fibration, Jones polynomial
##### Mathematical Subject Classification 2010
Primary: 57M25, 57M27, 57M50