Uniqueness of Seifert graphs

Question

If we make the bands and disks of a Seifert surface really small and really thin the surface collapses to a graph. It is called a Seifert graph.

1. If it is not a directed and weighted graph, can we say that for every unique Seifert graph there is a Seifert surface and vice versa?
2. If it is not a directed but weighted graph, can we say that for every unique Seifert graph there is a Seifert surface and vice versa?
3. If it is a both directed and weighted graph, can we say that for every unique Seifert graph there is a Seifert surface and vice versa?

Answer 1

With regards to your first question, I will assume that you are taking your Seifert graph to be embedded in $\mathbb R^3$. This can correspond to multiple Seifert surfaces by taking any Seifert surface with this graph and then adding lots of full twists into one of the bands. I am not sure what you expect the weights and directions to mean for the Seifert surface, so I am not sure how to respond to questions 2 and 3.