The Last Three T-degrees in Triply-Graded Link Homology

TL;DR

This paper studies the structure of triply graded link homology in specific degrees for links from positive and negative braids, providing explicit computations and revealing patterns of vanishing and uniformity.

Contribution

It introduces a diagrammatic approach to compute reduced triply graded link homology in top/bottom three degrees for braid closures, highlighting new structural insights.

Findings

Homology often vanishes in negative braid cases.

Homology exhibits striking uniformity in studied degrees.

Explicit $R$-module computations are provided.

Abstract

We investigate the structure of reduced triply graded link homology $\overline{\mathrm{HHH}}$ in the top/bottom three $T-$degrees for links arising as closures of positive/negative braids. Using a diagrammatic approach to the Hochschild cohomology of Soergel bimodules, we provide explicit computations of $\overline{\mathrm{HHH}}$ as $R-$modules in these degrees. Our results reveal that the homology here is often zero, especially in the negative braid case, and display striking uniformity.