I was wondering if there are any known examples of knots $K$ in $S^3$ with Seifert genus $g$ so that the lift of $K$ sitting inside its $n$-fold cyclic branched cover bounds an embedded surface of genus less than $g$?
If $g_n(K)$ is the smallest possible genus of an embedded surface bounding the lift of $K$ in its $n$-fold cyclic branched cover, what happens to $g_n(K)$ if we allow $n$ to grow? (Assume $K$ is not fibered.)
A non-trivial (and much more general) result of Gabai implies that $g_n(K)=g_1(K)$ for all $n$. This is encapsulated in the phrase ``Gromov norm equals Thurston norm". Roughly, the Gromov norm represents the minimal genus of an immersed Seifert surface, whereas the Thurston norm represents the minimal genus embedded Seifert surface. It follows that the minimal genus embedded Seifert surface realizes the minimal genus over all immersed Seifert surfaces.
Gabai's proof is interesting and enlightening. He constructs a taut foliation of the knot complement with a minimal genus Seifert surface as a leaf, making use of the theory of taut sutured manifolds that he develops in the papers. The euler class of this foliation gives a lower bound on the genus of immersed Seifert surfaces, and realized by the embedded Seifert surface.
