Immersed concordances of links and Heegaard Floer homology

TL;DR

This paper introduces a new approach using immersed concordances and Heegaard Floer homology to derive inequalities and criteria for link invariants, including the splitting number, with applications to L-space links and singularity deformations.

Contribution

It constructs a smooth 4D cobordism from immersed concordances and develops $d$-invariant inequalities to analyze link Floer homology, providing new bounds and criteria.

Findings

Established inequalities between $H$-functions of links.

Provided a Heegaard Floer criterion for bounding the splitting number.

Constructed infinite families of L-space links with large splitting numbers.

Abstract

An immersed concordance between two links is a concordance with possible self-intersections. Given an immersed concordance we construct a smooth four-dimensional cobordism between surgeries on links. By applying $d$-invariant inequalities for this cobordism we obtain inequalities between the $H$-functions of links, which can be extracted from the link Floer homology package. As an application we show a Heegaard Floer theoretical criterion for bounding the splitting number of links. The criterion is especially effective for L-space links, and we present an infinite family of L-space links with vanishing linking numbers and arbitrary large splitting numbers. We also show a semicontinuity of the $H$-function under $\delta$-constant deformations of singularities with many branches.