Let $K$ be a knot with genus $g$. Seifert's algorithm produces a surface of genus $k$ whose boundary is $K$. In general $k$ may be larger than $g$, but are there any bounds on how much larger it can be? That is, does there exist a function $\phi \colon \mathbb{N} \rightarrow \mathbb{N}$ such that $k \leq \phi(g)$?

## 1 Answer

Just a note about terminology: the minimal genus of a Seifert surface arising from Seifert's algorithm to a diagram of $K$ is called the *canonical genus* of $K$.

This paper by Brittenham and Jensen proves that the Whitehead double of an alternating pretzel knot $K$ has canonical genus equal to the crossing number of $K$.
Since Whitehead doubles have genus 1 and alternating pretzel knots have arbitrarily large crossing number, this shows that you cannot even define $\phi$ for $g=1$.
(In fact, there was an earlier paper of Tripp, *The canonical genus of Whitehead doubles of a family torus knots*, proving the same statement for the torus knots $T(2,2n+1)$.)

About the paper: the upper bound is given by applying Seifert's algorithm for a specific diagram, while the lower bound is given by Morton's inequality giving a lower bound on the canonical genus from the HOMFLY polynomial.