Four genera of links and Heegaard Floer homology

TL;DR

This paper explores the relationship between the genera of disjoint surfaces bounded by links with zero pairwise linking numbers and the $h$-function from Heegaard Floer homology, providing bounds and explicit calculations for $L$-space links.

Contribution

It introduces a method to determine the genera of surfaces bounded by links using the $h$-function, linking Heegaard Floer invariants to geometric properties of links.

Findings

The $h$-function provides lower bounds for the 4-genus of links.

For $L$-space links, the $h$-function is explicitly determined by Alexander polynomials.

Some $L$-space links achieve these bounds exactly.

Abstract

For links with vanishing pairwise linking numbers, the link components bound pairwise disjoint surfaces in $B^{4}$. In this paper, we describe the set of genera of such surfaces in terms of the $h$-function, which is a link invariant from Heegaard Floer homology. In particular, we use the $h$-function to give lower bounds for the 4-genus of the link. For $L$-space links, the $h$-function is explicitly determined by Alexander polynomials of the link and sublinks. We show some $L$-space links where the lower bounds are sharp, and also describe all possible genera of disjoint surfaces bounded by such links.