Khovanov-Rozansky homology and Directed Cycles

TL;DR

This paper introduces algebraic methods based on Khovanov-Rozansky homology to analyze directed graphs, enabling detection of cycles and determining the cycle packing number through elementary projective algebraic geometry.

Contribution

It applies Khovanov-Rozansky homology to graph theory, providing new algebraic techniques for cycle detection and cycle packing number calculation in directed graphs.

Findings

Cycle packing number determined via algebraic geometry.

Algebraic methods detect cycles containing specific vertices or edges.

Khovanov-Rozansky homology applied to directed graph analysis.

Abstract

We determine the cycle packing number of a directed graph using elementary projective algebraic geometry. Our idea is rooted in the Khovanov-Rozansky theory. In fact, using the Khovanov-Rozansky homology of a graph, we also obtain algebraic methods of detecting directed and undirected cycles containing a particular vertex or edge.