Computing Khovanov-Rozansky homology and defect fusion

TL;DR

This paper presents an algorithm and implementation for computing Khovanov-Rozansky homology, enabling the calculation of categorified sl(N) link invariants for various links and N values.

Contribution

The authors develop a new algorithm to reduce tensor products of matrix factorizations to finite rank and implement it in Singular, facilitating practical computations of link invariants.

Findings

Successfully computed Khovanov-Rozansky homology for multiple links

Implemented an efficient algorithm in Singular for tensor product reduction

Enabled calculations of invariants for various N values

Abstract

We compute the categorified sl(N) link invariants as defined by Khovanov and Rozansky, for various links and values of N. This is made tractable by an algorithm for reducing tensor products of matrix factorisations to finite rank, which we implement in the computer algebra package Singular.