Efficient calculations of $S$-invariants for links

TL;DR

This paper introduces an efficient algorithm for computing the $s$-invariant of links, demonstrating its effectiveness and extending applicability to $sl(3)$-link homology, surpassing previous spectral sequence methods.

Contribution

The paper presents a novel algorithm for calculating the $s$-invariant of links, including those from $sl(3)$-link homology, showing it outperforms spectral sequence-based approaches.

Findings

Effective computation of $s$-invariants for links.

Demonstrates limitations of spectral sequence methods.

Extends methods to $sl(3)$-link homology.

Abstract

We describe an algorithm that can effectively calculate the $s$-invariant of a link as defined by Beliakova and Wehrli. Our computations show that this cannot be done by merely calculating the $E_\infty$-page of the Bar-Natan--Lee--Turner spectral sequence. Our methods also work for $s$-invariants coming from sl(3)-link homology.