The number of surfaces of fixed genus in an alternating link complement

Let $L$ be a prime alternating link with $n$ crossings. We show that for each fixed $g$, the number of genus $g$ incompressible surfaces in the complement of $L$ is bounded by a polynomial in $n$. Previous bounds were exponential in $n$.
Comment: 9 pages, 2 figures, to appear in International Mathematics Research Notices