Matrix Factorizations and Kauffman Homology

TL;DR

This paper explores extending matrix factorization methods to define a new triply graded Kauffman homology, broadening the understanding of knot invariants beyond the sl(N) case.

Contribution

It introduces a novel triply graded theory categorifying the Kauffman polynomial and predicts Kauffman homology for simple knots, expanding the framework of knot homology.

Findings

Proposed a new triply graded Kauffman homology theory.

Tested the theory on simple knots.

Predicted Kauffman homology for several knots.

Abstract

The topological string interpretation of homological knot invariants has led to several insights into the structure of the theory in the case of sl(N). We study possible extensions of the matrix factorization approach to knot homology for other Lie groups and representations. In particular, we introduce a new triply graded theory categorifying the Kauffman polynomial, test it, and predict the Kauffman homology for several simple knots.