On the Seifert graphs of a link diagram and its parallels

TL;DR

This paper characterizes Seifert graphs derived from link diagrams using Eulerian subgraphs and explores how their genus changes when forming parallels of the link diagram.

Contribution

It provides a new characterization of Seifert graphs via Eulerian subgraphs and analyzes the genus relationship for parallels of link diagrams.

Findings

Seifert graphs are bipartite and can be characterized by Eulerian subgraphs.

The genus of ribbon graphs of parallels relates systematically to the original diagram.

The paper extends understanding of the topological properties of link diagram parallels.

Abstract

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.