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

TL;DR

This paper proves that for non-split prime alternating links, the number of genus-$g$ Seifert surfaces is polynomially bounded in the number of crossings, improving upon previous exponential bounds.

Contribution

It establishes explicit polynomial bounds on the number of genus-$g$ Seifert surfaces for certain links, advancing understanding of their topological complexity.

Findings

Number of genus-$g$ Seifert surfaces is polynomially bounded in crossings

Bound applies to all spanning surfaces with fixed Euler characteristic

Improves previous exponential bounds

Abstract

Let $L$ be a non-split prime alternating link with $n>0$ crossings. We show that for each fixed $g$, the number of genus-$g$ Seifert surfaces for $L$ is bounded by an explicitly given polynomial in $n$. The result also holds for all spanning surfaces of fixed Euler characteristic. Previously known bounds were exponential.