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On the Seifert graphs of a link diagram and its parallels

Published online by Cambridge University Press:  20 March 2012

STEPHEN HUGGETT ,
IAIN MOFFATT  and
NATALIA VIRDEE
STEPHEN HUGGETT
Affiliation:
School of Computing and Mathematics, University of Plymouth, PL4 8AA, Devon. e-mail: s.huggett@plymouth.ac.uk
IAIN MOFFATT
Affiliation:
Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688, U.S.A. e-mail: imoffatt@jaguar1.usouthal.edu
NATALIA VIRDEE
Affiliation:
School of Computing and Mathematics, University of Plymouth, PL4 8AA, Devon. e-mail: natalia.virdee@plymouth.ac.uk
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Abstract

Recently, Dasbach, Futer, Kalfagianni, Lin and Stoltzfus extended the notion of a Tait graph by associating a set of ribbon graphs (or, equivalently, cellularly embedded graphs) to a link diagram. Here we focus on Seifert graphs, which are the ribbon graphs of a knot or link diagram that arise from Seifert states. We provide a characterization of Seifert graphs in terms of Eulerian subgraphs. This characterization can be viewed as a refinement of the fact that Seifert graphs are bipartite. We go on to examine the family of ribbon graphs that arises by forming the parallels of a link diagram and determine how the genus of the ribbon graph of a r-fold parallel of a link diagram is related to that of the original link diagram.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 153 , Issue 1 , July 2012 , pp. 123 - 145
Copyright
Copyright © Cambridge Philosophical Society 2012

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