Heegaard Floer homology of (n,n)-torus links: computations and questions

TL;DR

This paper investigates the Heegaard Floer homology of (n,n)-torus links, computing ranks and gradings in many cases, and proposes a conjecture for the complete structure, highlighting open questions in the field.

Contribution

It provides detailed computations of the link homology for (n,n)-torus links and introduces a conjecture for its full description, advancing understanding in link homology.

Findings

Identified the Alexander multigradings supporting non-trivial homology as a string of n-1 unit hypercubes.

Computed ranks and gradings of homology in nearly all Alexander gradings.

Proposed a conjecture for the complete description of the link homology.

Abstract

In this article we study the Heegaard Floer link homology of $(n, n)$-torus links. The Alexander multigradings which support non-trivial homology form a string of $n-1$ unit hypercubes in $\mathbb{R}^{n}$, and we compute the ranks and gradings of the homology in nearly all Alexander gradings. We also conjecture a complete description of the link homology and provide some support for this conjecture. This article is taken from the author's 2007 Ph.D. thesis and contains several open questions.