Upper bound on the genus produced by Seifert's algorithm

Question

Given a knot $K$ we can apply Seifert's algorithm to produce a surface whose boundary is $K$. The genus of this surface is not necessarily minimal. Is there an upper bound on the genus of the surface produced by Seifert's algorithm in terms of the knot genus of $K$? That is, if $K$ has genus $g$ and Seifert's algorithm produces a surface of genus $n$ does there exist a function $\phi \colon \mathbb{N} \rightarrow \mathbb{N}$ such that $n \leq \phi(g)$?

Answer 1

First, let me answer the question you probably didn't mean: is there a function $\phi:\mathbb{N}\to\mathbb{N}$ such that, given a knot diagram $D$ of a knot $K$ of genus $g$, the genus $n$ of the surface obtained from Seifert's algorithm for $D$ satisfies $n\leq \phi(g)$? The answer is "no" since you can make diagrams of the unknot whose corresponding surface's genus can be made arbitrarily large. For example, the following unknot diagram's surface from Seifert's algorithm has genus $4$:

Extending the pattern horizontally increases the genus of the corresponding surface. (I checked this with KnotFolio. The genus of the Seifert's algorithm surface is under "Can. genus" in the "Diagram information" section. The program does not give the true canonical genus, so it's arguably mislabeled.)

The stronger interpretation of your question is about canonical genus, the minimal genus over all diagrams of $K$ of the surface obtained through Seifert's algorithm. It's not immediately clear to me that canonical genus has an algorithm, so I hesitate to call the canonical genus "the genus of the surface obtained by Seifert's algorithm," which is why I'm giving two answers. When answering this question, I came across a paper that says the answer is "no," since it purports to show that the difference between canonical genus and the knot genus can be arbitrarily large. (I say "purports" because I've only read the abstract.) I don't know whether this is the first paper with this result, since, from the abstract, it seems like their contribution is that they introduce an intermediate quantity called "free genus."

Kobayashi, Masako; Kobayashi, Tsuyoshi, On canonical genus and free genus of knot, J. Knot Theory Ramifications 5, No. 1, 77-85 (1996). ZBL0859.57009.