Structures in HOMFLY-PT homology

TL;DR

This paper investigates the detailed structure of HOMFLY-PT homology, utilizing recent computational data and algebraic actions to analyze knot invariants and their relationships.

Contribution

It introduces a new analysis of HOMFLY-PT homology structures using recent computational data and algebraic actions, and compares different knot invariants.

Findings

Computed HOMFLY-PT S-invariant for all knots in the dataset.

Compared HOMFLY-PT invariants with $rak{sl}(N)$ concordance invariants.

Identified structural patterns in triply graded homology.

Abstract

We study the structure of triply graded Khovanov-Rozansky homology using both the data recently computed by Nakagane and Sano for knots up to 11 crossings, and the $\mathfrak{sl}(2)$ action defined by the second author, Hogancamp and Mellit. In particular, we compute the HOMFLY-PT $S$-invariant for all knots in the dataset, and compare it to the $\mathfrak{sl}(N)$ concordance invariants.