State graphs and fibered state surfaces

Girão, Darlan; Purcell, Jessica

doi:10.2140/agt.2020.20.987

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State graphs and fibered state surfaces

Darlan Girão and Jessica S Purcell

Algebraic & Geometric Topology 20 (2020) 987–1014

DOI: 10.2140/agt.2020.20.987

Abstract

Associated to every state surface for a knot or link is a state graph, which embeds as a spine of the state surface. A state graph can be decomposed along cut-vertices into graphs with induced planar embeddings. Associated with each such planar graph is a checkerboard surface, and each state surface is a fiber if and only if all of its associated checkerboard surfaces are fibers. We give an algebraic condition that characterizes which checkerboard surfaces are fibers directly from their state graphs. We use this to classify fibering of checkerboard surfaces for several families of planar graphs, including those associated with $2$ –bridge links. This characterizes fibering for many families of state surfaces.

Keywords

fibered links, Kauffman state, state graph, state surface, $2$–bridge link

Mathematical Subject Classification 2010

Primary: 57M25, 57M27

References

Publication

Received: 18 February 2019

Revised: 16 June 2019

Accepted: 8 July 2019

Published: 23 April 2020

Authors

Darlan Girão
Department of Mathematics Universidade Federal do Ceará Fortaleza-CE Brazil

Jessica S Purcell
School of Mathematical Sciences Monash University Clayton, VIC Australia

Volume 20, issue 2 (2020)