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Let $L$ be a non-split prime alternating link with $n>0$ crossings. We showthat for each fixed $g$, the number of genus-$g$ Seifert surfaces for $L$ isbounded by an explicitly given polynomial in $n$. The result also holds for allspanning surfaces of fixed Euler characteristic. Previously known bounds wereexponential.

Alternating links have at most polynomially many Seifert surfaces of fixed genus