Rational genus and Heegaard Floer homology

TL;DR

This paper explores the relationship between rational genus and Heegaard Floer homology, providing a new proof of existing bounds and extending the concepts to rational slice genus in product 4-manifolds.

Contribution

It offers a simplified proof of known bounds on rational genus using Heegaard Floer invariants and introduces a rational slice genus concept with similar minimization properties.

Findings

Established a lower bound for rational genus via the $d$-invariant.

Identified Floer simple knots as genus minimizers.

Extended the genus minimization results to rational slice genus in $Y\times I$."

Abstract

Turaev defined a function on the first homology of a rational homology 3-sphere $Y$ as the minimal rational Seifert genus of all knots in this homology class. Ni and the first author discovered a lower bound of this function using the Heegaard Floer $d$-invariant and showed that Floer simple knots are rational Seifert genus minimizers. In this paper, we give a simple reproof of the above results. We then define a version of rational slice genus for knots in the product 4-manifold $Y\times I$ and investigate the analogous minimal genus problem. We prove the same lower bound in terms of the $d$-invariant formula and the same genus minimizers given by Floer simple knots.