MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.)

share|cite|improve this question
up vote 14 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.