A higher-order genus invariant and knot Floer homology

TL;DR

This paper explores the limitations of knot Floer homology in detecting certain invariants of minimal genus Seifert surfaces, introducing a new invariant for genus one knots and discussing bounds from metabelian $L^2$-signatures.

Contribution

It defines a new invariant for algebraically slice, genus one knots and demonstrates that knot Floer homology does not detect this invariant.

Findings

Knot Floer homology does not detect the new invariant.

A new invariant for genus one knots is introduced.

Metabelian $L^2$-signatures provide lower bounds for the invariant.

Abstract

It is known that knot Floer homology detects the genus and Alexander polynomial of a knot. We investigate whether knot Floer homology of $K$ detects more structure of minimal genus Seifert surfaces for $K$. We define an invariant of algebraically slice, genus one knots and provide examples to show that knot Floer homology does not detect this invariant. Finally, we remark that certain metabelian $L^2$-signatures bound this invariant from below.