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

TL;DR

This paper proves that for prime alternating links, the number of genus g incompressible surfaces in their complements grows at most polynomially with the number of crossings, improving previous exponential bounds.

Contribution

It establishes a polynomial bound on the number of genus g incompressible surfaces in prime alternating link complements, a significant improvement over prior exponential bounds.

Findings

Number of genus g surfaces is polynomially bounded in n

Previous bounds were exponential in n

Results apply to prime alternating links

Abstract

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 $n$.