Fiber detection for state surfaces

Futer, David

doi:10.2140/agt.2013.13.2799

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Fiber detection for state surfaces

David Futer

Algebraic & Geometric Topology 13 (2013) 2799–2807

DOI: 10.2140/agt.2013.13.2799

Abstract

Every Kauffman state $σ$ of a link diagram $D (K)$ naturally defines a state surface $S_{σ}$ whose boundary is $K$ . For a homogeneous state $σ$ , we show that $K$ is a fibered link with fiber surface $S_{σ}$ 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.

Keywords

adequate knot, homogeneous knot, spanning surface, fibration, Jones polynomial

Mathematical Subject Classification 2010

Primary: 57M25, 57M27, 57M50

References

Publication

Received: 3 May 2012

Revised: 27 March 2013

Accepted: 15 April 2013

Published: 18 July 2013

Authors

David Futer
Department of Mathematics Temple University Philadelphia, PA 19122 USA
http://math.temple.edu/~dfuter

Volume 13, issue 5 (2013)