Links and Graphs

Abstract

In this thesis we derive some basic properties of graphs G embedded in a surface determining a link diagram D(G), having a specified number μ(D(G)) of components. ( The relationship between the graph and the link diagram comes from the tangle which replaces each edge of the graph). Firstly, we prove that μ (D(G)) ≤ f (G) + 2g, where f (G) is the number of faces in the embedding of G and g is the genus of the surface. Then we focus on the extremal case, where μ (D(G)) = f (G) + 2g. We note that μ (D(G)) does not change when undergoing graph Reidemeister moves or embedded ∆ ↔ Y exchanges. It is also useful that μ(D(G)) changes only very slightly when an edge is added to the graph. We finish with some observations on other possible values of μ(D(G)). We comment on two cases: when μ = 1, and the Petersen and Heawood families of graphs. These two families are obtained from K6 and K7 respectively by using ∆ ↔ Y exchanges.