The concordance invariant tau in link grid homology

TL;DR

This paper generalizes the tau-invariant to links using filtered link grid homology, demonstrating its invariance under concordance, its lower bound on slice genus, and applications to Legendrian links.

Contribution

It introduces a new link invariant extending tau, proves its concordance invariance, and applies it to bound slice genus and Legendrian link invariants.

Findings

Invariant remains unchanged under strong concordance.

Provides sharp lower bounds for slice genus of torus links.

Applications to Legendrian link invariants in contact 3-sphere.

Abstract

We introduce a generalization of the Ozsv\'ath-Szab\'o $\tau$-invariant to links by studying a filtered version of link grid homology. We prove that this invariant remains unchanged under strong concordance and we show that it produces a lower bound for the slice genus of a link. We show that this bound is sharp for torus links and we also give an application to Legendrian link invariants in the standard contact 3-sphere.