Seifert surfaces distinguished by sutured Floer homology but not its Euler characteristic

TL;DR

This paper introduces a novel application of sutured Floer homology to distinguish non-isotopic Seifert surfaces for knots with trivial Alexander polynomial, surpassing traditional invariants like Euler characteristic.

Contribution

It demonstrates the first use of sutured Floer homology, specifically Longitude Floer Homology, to differentiate Seifert surfaces, providing new tools for knot and surface classification.

Findings

Sutured Floer homology can distinguish Seifert surfaces where Euler characteristic cannot.

Constructed examples of knots with non-isotopic Seifert surfaces sharing the same Alexander polynomial.

Introduced a technique using Longitude Floer Homology to simplify computations.

Abstract

In this paper we find a family of knots with trivial Alexander polynomial, and construct two non-isotopic Seifert surfaces for each member in our family. In order to distinguish the surfaces we study the sutured Floer homology invariants of the sutured manifolds obtained by cutting the knot complements along the Seifert surfaces. Our examples provide the first use of sutured Floer homology, and not merely its Euler characteristic(a classical torsion), to distinguish Seifert surfaces. Our technique uses a version of Floer homology, called Longitude Floer Homology in a way that enables us to bypass the computations related to the SFH of the complement of a Seifert surface.