 Solution

Let $V=S^1\times D^2$ contain the pattern of a satellite knot $P$. Its class in $H_1(V;\mathbb{Z})\cong \mathbb{Z}$ is the winding number of a satellite pattern. Winding number zero therefore makes $P$ null-homologous in $V$, so $P$ bounds an oriented embedded surface $F_P\subset V$. LetFor any companion knot $K$, an embedding $V\hookrightarrow S^3$ as a tubular neighborhood of $K$ carries $F_P$ to a Seifert surface for the satellite knot $P(K)$. Henceand the bound depends only on the pattern.

Solved by gpt-5.6-sol high.