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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?

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If $K$ is a non-trivial knot, then $\chi(S)<0$, so $S$ admits a hyperbolic structure as a surface. But in general, that metric does not arise from the emdedding into $S^3\setminus K$.

If $S^3\setminus K$ is hyperbolic, and $S$ is a properly embedded $\pi_1$-essential surface in $S^3\setminus K$, then $S$ is either virtually fibered, accidental, or quasi-fuchsian. See Bonahon, or Canary, Epstein, Green - Notes on notes of Thurston, or Thurston's notes.

In the case that $S$ is a minimal genus Seifert surface, then $S$ is either a fiber, or $S$ is quasi-fuchsian by work of Fenley.

Within the class of quasi-fuchsian surfaces are the surfaces which are totally geodesic. Totally geodesic Seifert surfaces will inherit a metric of constant curvature $-1$. This is quite rare for a Seifert surface, but examples are known due to Adams and Schoenfeld.

Perhaps the best bet for a canonical geometry on a Seifert surface $S$ in a hyperbolic knot complement, is to give $S$ the structure of a pleated surface. Again see Thurston's notes.

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    $\begingroup$ Yes, I would have mentioned the pleated surfaces which are the two boundary components of the convex core (but of course these are immersed). Choosing orientations, one could pick one of these out canonically. One can also pick out a canonical minimal representative of a quasifuchsian surface by using the lattice ordering on minimal surfaces. In the fibered case, there's the singular solv metric which is canonical in some sense (a scale parameter can be chosen so that the volume is 1). In this case, there's a canonical singular Lorentz metric on the surface up to Lorentzian isometry! $\endgroup$
    – Ian Agol
    Oct 27, 2020 at 22:26

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