Sutured Floer homology distinguishes between Seifert surfaces

TL;DR

This paper demonstrates that sutured Floer homology can distinguish between different minimal genus Seifert surfaces for the same knot, providing new tools for knot surface analysis.

Contribution

It presents the first example of knots with multiple minimal genus Seifert surfaces distinguished by sutured Floer homology and introduces an infinite family of such examples.

Findings

Sutured Floer homology distinguishes between Seifert surfaces.

The Euler characteristic of sutured Floer homology can differentiate surfaces.

An infinite family of knots with distinguishable Seifert surfaces is constructed.

Abstract

We exhibit the first example of a knot in the three-sphere with a pair of minimal genus Seifert surfaces that can be distinguished using the sutured Floer homology of their complementary manifolds together with the Spin^c-grading. This answers a question of Juh\'asz. More precisely, we show that the Euler characteristic of the sutured Floer homology of the complementary manifolds distinguishes between the two surfaces, as does the sutured Floer polytope introduced by Juh\'asz. Actually, we exhibit an infinite family of knots with pairs of Seifert surfaces that can be distinguished by the Euler characteristic.