Fiber detection for state surfaces

Abstract

Every Kauffman state σ of a link diagram D.K/ naturally defines a state surface Sσ whose boundary is K. For a homogeneous state σ, we show that K is a fibered link with fiber surface Sσ if and only if an associated graph G′ σ 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.