# Upper bounds on the genus of the surface produced by Seifert's algorithm

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 $$k \leq \phi(g)$$?

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 $$T(2,2n+1)$$.)