[Paper Review] Colored sl(N) link homology via matrix factorizations

Hao Wu|arXiv (Cornell University)|Oct 10, 2011

Geometric and Algebraic Topology28 references3 citations

TL;DR

This paper presents a categorification of the colored $ \mathfrak{sl}(N)$ link invariant for links colored by wedge powers of the defining representation using matrix factorizations, generalizing Khovanov-Rozansky homology. It constructs a triply graded homology theory via chain complexes of graded matrix factorizations that is invariant under Reidemeister moves and recovers the original Khovanov-Rozansky construction when all components are uncolored (color 1).

ABSTRACT

The Reshetikhin-Turaev sl(N) polynomial of links colored by wedge powers of the defining representation has been categorified via several different approaches. Here, we give a concise introduction to the categorification using matrix factorizations, which is a direct generalization of the Khovanov-Rozansky homology. Full details of the construction are given in [arXiv:0907.0695]. We also briefly review deformations and applications of this categorification given in [arXiv:1002.2662, arXiv:1011.2254, arXiv:1102.0586].

Motivation & Objective

• To provide a concise introduction to the categorification of the Reshetikhin-Turaev $ \mathfrak{sl}(N)$ polynomial for links colored by wedge powers of the defining representation using matrix factorizations.
• To generalize the Khovanov-Rozansky homology construction to include colored links (not just uncolored ones) via matrix factorizations.
• To establish the invariance of the resulting homology under Reidemeister moves, ensuring it defines a link invariant.
• To clarify the relationship between the new construction and the original Khovanov-Rozansky homology in the uncolored case.
• To summarize key technical tools and results from prior work, including deformations and applications, in a self-contained manner.

Proposed method

• Constructs a chain complex $C(D)$ of graded matrix factorizations associated to a knotted MOY graph $D$, using markings and algebraic data from symmetric polynomials.
• Employs the MOY calculus—graphical relations for evaluating the Reshetikhin-Turaev $ \mathfrak{sl}(N)$ polynomial—within the framework of matrix factorizations.
• Defines a triply graded complex with $ \mathbb{Z}_2$, quantum, and homological gradings, generalizing the grading structure of Khovanov-Rozansky homology.
• Uses morphisms induced by local moves (e.g., fork sliding, circle creation/annihilation, edge splitting/merging) to establish invariance under Reidemeister moves.
• Applies the Krull-Schmidt property and homotopy theory of matrix factorizations to analyze the structure of the chain complex.
• Takes the homology of the complex first with respect to the matrix factorization differential, then with respect to the chain complex differential, to define the final homology $H(D)$.

Experimental results

Research questions

• RQ1 How can the Reshetikhin-Turaev $ \mathfrak{sl}(N)$ polynomial for links colored by wedge powers of the defining representation be categorified using matrix factorizations?
• RQ2 What is the structure of the resulting homology theory, and how does it generalize the Khovanov-Rozansky homology for uncolored links?
• RQ3 How are the invariance properties of the homology under Reidemeister moves established in this matrix factorization framework?
• RQ4 What is the relationship between the new construction and the original Khovanov-Rozansky homology when all components are colored by 1?
• RQ5 How do deformations and applications of this categorification extend the scope of the theory beyond the standard case?

Key findings

• The chain complex $C(D)$ associated to a tangle diagram $D$ is a bounded complex over the homotopy category of graded matrix factorizations and carries a $ \mathbb{Z}_2 \oplus \mathbb{Z} \oplus \mathbb{Z}$-grading.
• The homotopy type of $C(D)$ is invariant under Reidemeister moves, ensuring that the resulting homology $H(D)$ is a link invariant.
• When all components of $D$ are colored by 1, the construction recovers the original Khovanov-Rozansky chain complex, confirming consistency with the uncolored case.
• The Euler characteristic of the homology $H(D)$ recovers the Reshetikhin-Turaev $ \mathfrak{sl}(N)$ polynomial, verifying categorification.
• The construction uses a recursive framework based on MOY relations and local moves, with morphisms induced by graphical changes (e.g., saddle moves, circle creation) ensuring invariance.
• The final homology $H(D)$ is defined by first taking homology with respect to the matrix factorization differential $d_{mf}$, then with respect to the chain complex differential $d^*$, yielding a triply graded invariant.