On the Seifert graphs of a link diagram and its parallels

Details On the Seifert graphs of a link diagram and its parallels On the Seifert graphs of a link diagram and its parallels

2011-06-21

arxiv.org/abs/1106.4197v1

Mathematics

Geometric Topology

Scientific paper

10.1017/S0305004112000102

Recently, Dasbach, Futer, Kalfagianni, Lin, and Stoltzfus extended the notion of a Tait graph by associating a set of ribbon graphs (or equivalently, 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.

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