I originally posted this on stackexchange, but it hasn't gotten an answer. I hope it's not inappropriate for this forum.
Suppose I have a knot $K: S^1 \hookrightarrow S^3$ with minimal genus Seifert surface $S$. I would like to know whether we can endow $S$ with a geometry that is independent somehow of our embedding, assuming said embedding satisfies whatever conditions are appropriate.
I understand this is a naive/vague question, but I don't have any real background in geometric topology and so I'm not sure what conditions we would want the embedding to satisfy to even begin looking at geometric properties as knot invariants. I imagine there are some elementary theorems for 2-manifolds with boundary that would be useful here, but I'm not sure where to look for them.
I know that hyperbolic knots are characterized by the fact that their complements can be endowed with a geometry having constant curvature $-1$. Since we can embed $S$ in the complement as a smooth submanifold, does this also mean that all smooth Seifert surfaces for hyperbolic knots can likewise be given a geometry with constant curvature $-1$? Are we able to say anything at all about the surfaces for torus and satellite knots?